Dissertations / Theses on the topic 'Numerical'

Although only our species has achieved high level of mathematical reasoning, numerical abilities are not a human prerogative and in last decades comparative research showed that several animal species display rudimentary numerical capacities (Agrillo & Beran, 2013). The ability to discriminate between quantities provide multiple benefits in different ecological contexts. For instance, numerical abilities can be useful to select the larger amount of food (Hunt et al., 2008), to reduce the probability of being spotted by predators by getting protection within the largest group of social companions (Cresswell, 1994) and to decide whether attack another group based on the assessment of the relative number of intruders (Benson-Amram et al., 2011). In particular, the discovery in recent years that even simple organisms, such as fish, possess numerical abilities similar to primates has made possible the use of fish as an animal model to study numerical cognition in the absence of language. To date, different studies have shown that fish are able to select the larger shoal of conspecifics (Agrillo et al., 2008) and can be trained to discriminate between groups of figures differing in numerosity both when allowed to use number and continuous quantities and when only number was available (Agrillo et al., 2009, 2010). Fish can also make a spontaneous use of numerical information with apparently the same effort required to discriminate continuous quantities (Dadda et al., 2009). These abilities seem to be partially inborn as one-day old fish are already able to discriminate between small groups of peers (Bisazza et al., 2010). Nonetheless several questions about numerical abilities in fish are still unanswered. For instance, it is unclear whether numerical systems are the same among different species, whether numerical acuity may be affected by different factors, such as cooperation among individuals and the presence of items in motion or whether newborn fish could be trained to discriminate between sets of items. The aim of the present study was to fill this gap. In particular, the first part of the thesis deals with some of the open questions about numerical cognition in adult fish; the second part is focused on the ontogeny of numerical competence. In the first study (Section 4.1) we set up a novel procedure for training fish to discriminate between sets of stimuli (groups of geometrical figures) differing in numerosity as the previous methodology used to train fish was time-consuming, suitable only for social species and potentially stressful for fish. To validate the method, we replicated two published experiments that used operant conditioning to investigate mosquitofish (Gambusia holbrooki) abilities to discriminate between small sets of items and the influence of numerical ratio and total number of figures on large number discrimination (Agrillo et al., 2009, 2010). In the new procedure a pair of stimuli differing in numerosity was introduced at the opposite ends of the experimental tank and a food reward was released in correspondence of the stimulus to be reinforced. Fish were initially trained on an easy numerical ratio (0.5) and were then tested in non-reinforced probe trials for their ability to generalize to new numerosities. The new procedure designed replicated previous results: fish proved able to discriminate up to 2 vs. 3 figures and their performance in the large number range decreased while increasing the numerical ratio though their numerical acuity seemed to have no upper limit. In addition, the new method proved to be rapid, applicable to different fish species and efficient to study discrimination learning in fish in tasks requiring visual stimuli. As a consequence, the novel protocol was adopted in all the training experiments presented in this thesis. The second study (Section 4.2) focused on a potential limit in numerical cognition research: the lack of cross-species studies using the same methodology. The question of whether all vertebrates share the same numerical systems or rather numerical abilities have appeared multiple times during evolution in response to specific selective pressures imposed by the environment, represents one of the main issues of animal cognition. Despite the large number of published data, results are inconsistent because the methodologies adopted vary across studies, making difficult any inter-specific comparison. To date no study has investigated if different fish species have the same numerical systems. This experiment represents the first inter-specific study using the same methodology in fish. Five fish species as diverse as guppies (Poecilia reticulata), zebrafish (Danio rerio), angelfish (Pterophyllum scalare), redtail splitfin (Xenotoca eiseni) and Siamese fighting fish (Betta splendens) were trained on an easy numerical ratio (0.50) and then were compared in their ability to generalize to more difficult ratios (0.67 and 0.75), or to a larger (25 vs. 50) or a smaller (2 vs. 4) total set size. Results showed interesting similarities among the species, opening the possibility of shared numerical systems among phylogenetically distantly related species, more in accord with the existence of ancient quantification systems inherited from a common ancestor than with an independent evolution of numerical abilities in different species. Another important question in the study of numerical cognition concerns the influence of contextual factors on the numerical capacities of a species. It is possible that the performance observed in a numerical task is limited to the specific context in which such abilities are observed rather than reflecting the full numerical competence of a species. To this purpose, the third (Section 5.1) and fourth (Section 5.2) studies investigated the potential influence on fish numerical acuity of factors that normally occur in nature, namely, the cooperative behavior within group and the perception of figures in motion. In natural environment grouping animals interact with each other and these repeated interactions among individuals can affect adaptive response. Recent studies have provided evidence that, in some contexts, collective actions allow to bypass the cognitive limits of a species and to solve problems that go beyond the capacity of a single individual (Krause et al., 2010, Couzin, 2009). To date, all numerical studies in non-human animals have tested subjects individually and it is not known whether collective behavior can enhance the capacity to solve numerical tasks. The third study (Section 5.1) aimed to verify whether fish in dyads were more accurate than single individuals in two different numerical discrimination tasks. In the first task, guppies were required to join the larger group of conspecifics (4 vs. 6); in the second one fish were trained to discriminate between sets of figures (0.5 ratio) and hence were tested in discriminations of increasing difficulty (0.67 and 0.75 ratios). Results showed that dyads performed better than singletons in selecting the larger group of social companions and also made better numerical discriminations of arrays of dots, showing that collective behavior may yield benefits that go beyond the single ecological context. In addition, in both conditions, the better individual of the dyad spontaneously emerged as the leader. Interestingly, the results here obtained aligned with data collected in adult humans where dyadic performance was superior than individual in a collective enumeration task (Bahrami et al., 2013), thus suggesting that cooperation similarly increases numerical acuity in two distantly related species, such as humans and fish. The motion of items is another factor that might potentially affect numerical abilities. Animals are naturally exposed to moving items (e.g., prey, predators) and hence the movement represents a relevant cue in their life. It is known that fish ability to discriminate between small and large groups of conspecifics is differently affected by the quantity of movement of social companions (Agrillo et al., 2008). However it is still unexplored whether fish can discriminate between two-dimensional figures in motion and whether their accuracy is the same in the small and large number range. For example, it has been reported that adult humans are faster and more accurate in estimating small numerosities (≤ 4) of dynamic items than large numerosities (≥ 4), supporting the hypothesis of two distinct numerical systems (Trick et al., 2003, Alston & Humphreys, 2004). To this aim, in the fourth study (Section 5.2) guppies were trained (0.5 ratio) and tested (0.75 ratio) with either static or moving stimuli. We observed a similar effect of items in motion in fish: while a 0.75 ratio was not discriminated with static stimuli in either numerical range (3 vs. 4 and 9 vs. 12), guppies were able to discriminate this ratio with items in motion but only in the small number range (3 vs. 4). To date, comparative psychologists disagree as to whether in non-human species a single system accounts for discriminations over the whole numerical range (called “Approximate number system”), or a distinct system operates over the small number range (≤4) (called “Object tracking system”). Although the results do not represent a direct evidence for the existence of a separate system in the range 1-4, the differential effect of motion reported in guppies reinforces the idea of separate cognitive systems for small and large numbers, in line with data collected in humans. Despite no direct comparisons have been made between fish and humans in this thesis, the similarities between the two species are worth noting as they raise the intriguing possibility that the foundation of our numerical abilities might be evolutionarily more ancient than previously thought, dating back at least as far as the divergence between fish and land vertebrates. The second part of the thesis focused on the development of numerical abilities using newborn guppies as a model species. Developmental studies can provide useful insights with respect to the existence of a single or multiple systems of numerical representation. For instance, exploring developmental trajectories of numerical skills in different contexts can help us to assess whether the same or distinct numerical systems are used in different tasks. Since an adequate method to study discrimination learning in newborn guppies was not available, in the fifth study (Section 6.1) we designed a procedure by taking into account the social needs of young individuals in order to minimize potential stress due to social deprivation, without interfering with the normal development of their behavioral repertoire. We investigated the development of social behavior in the first two weeks of life by using a spontaneous choice task where newborn guppies could choose between social companions and an empty compartment. Then, newborns were given the choice between their own mirror image and a group of peers to assess whether mirrors could be used as a substitute for social companions during experiments. Based on the findings of these experiments, the protocol for discrimination learning in adult fish was adapted to study shape discrimination in newborn fish. Newborn guppies proved capable to learn a simple shape discrimination after few trials and the training method was then used in the last study (Section 6.2) to investigate their numerical competence using sets of two-dimensional objects, as commonly done with adult fish. At present only Bisazza and colleagues (2010) investigated the ontogeny of numerical abilities in fish. The authors found that, at birth, the capacity of guppies to discriminate between shoals differing by one individual included all numerical contrasts in the range 1-4; young guppies proved also be able to discriminate small numerosities by using numerical information only. In Section 6.2 we investigated whether newborns could be trained to discriminate between small sets of figures. To this purpose, we set up three different experimental conditions to study the influence of continuous quantities that co-vary with numbers (cumulative surface area, density, etc.). In the first one number and continuous quantities were simultaneously available, in the second condition only numerical information was available and in the last one, numerical information was made irrelevant (3 vs. 3) and only continuous quantities were available. The result that fish discriminated only very easy numerical contrasts in the range 1-4 when both number and continuous variables were available was in contrast with the results of shoal discrimination experiments (Bisazza et al., 2010) thus suggesting that newborns’ capacity to use number is specific to social stimuli. On the whole data on guppies, both adult fish and newborns, are suggestive of the existence of multiple quantification mechanisms in fish which are domain-specific and serve to solve a limited set of problems in accordance with the hypothesis proposed by different authors (Feigenson et al., 2004; Spelke, 2000). In sum, the data collected in this thesis indicate that even fish, which are provided with a much smaller brain than warm-blooded vertebrates, can discriminate between quantities and solve complex numerical tasks, in line with evidence in other research fields which suggest that processing numerical information might not require complex neural circuits (Hope et al., 2010). This goes together with recent discovery that bony fish possess several other cognitive abilities that were previously believed to be uniquely present in species provided with large, complex brains (i.e. mammalian and avian species) (Bshary et al., 2002). For all these reasons, fish may become a proper model to study cognitive abilities and in particular numerical competence.
Sebbene solamente la nostra specie abbia raggiunto un elevato livello di competenze matematiche, le capacità numeriche non sono una prerogativa umana e negli ultimi decenni la ricerca comparata ha documentato come molte specie animali posseggano rudimentali abilità numeriche (Agrillo & Beran, 2013). La capacità di saper discriminare tra diverse quantità risulta essere vantaggiosa in diversi contesti ecologici. Per esempio, tale abilità può essere utile per scegliere la quantità maggiore di cibo (Hunt et al., 2008), per ridurre la probabilità di essere predati - ottenendo protezione dal gruppo di conspecifici più numeroso (Cresswell, 1994) - e per decidere se intraprendere interazioni aggressive contro un altro gruppo in base al numero di potenziali rivali (Benson-Amram et al., 2011). In particolare, la recente scoperta che persino organismi semplici, come i pesci, posseggono abilità numeriche simili a quelle osservate nei primati ha reso possibile l'utilizzo dei pesci come modello animale per studiare la cognizione numerica in assenza del linguaggio. Ad oggi, diversi studi hanno infatti dimostrato che i pesci sono capaci di selezionare il gruppo di conspecifici più numeroso (Agrillo et al., 2008) e possono essere addestrati a discriminare tra gruppi di figure di diversa numerosità, sia quando possono utilizzare l’informazione numerica e le variabili continue simultaneamente, sia nel caso in cui sia disponibile solamente l’informazione numerica (Agrillo et al., 2009, 2010). È stato inoltre dimostrato che i pesci sono in grado di discriminare tra quantità usando spontaneamente il numero, apparentemente con lo stesso sforzo cognitive richiesto per discriminare le variabili continue (Dadda et al., 2009). Queste capacità sembrano essere in parte innate, dal momento che gli avannotti di un giorno di vita sono già in grado di discriminare tra piccoli gruppi di conspecifici (Bisazza et al., 2010). Tuttavia diverse domande sulle abilità numeriche nei pesci sono ancora senza risposta. Ad esempio, non è chiaro se i sistemi numerici siano gli stessi fra specie differenti, se l'acuità numerica possa essere influenzata da diversi fattori, come la cooperazione tra gli individui e la presenza di oggetti in movimento o se i pesci appena nati possano essere addestrati a discriminare tra gruppi di oggetti bidimensionali. Lo scopo della presente tesi è stato pertanto quello di colmare queste lacune. In particolare, la prima parte della tesi affronta alcune delle questioni aperte sulla cognizione numerica nei pesci adulti, mentre la seconda parte è focalizzata sull’ontogenesi delle abilità numeriche. Nel primo lavoro (Sezione 4.1) è stata messa a punto una nuova procedura per addestrare i pesci a discriminare tra stimoli bidimensionali (gruppi di figure geometriche) di diversa numerosità, dal momento che il metodo precedentemente utilizzato in letteratura richiedeva tempi prolungati, era adatto solo per le specie sociali ed era potenzialmente stressante per i pesci. Per verificare la validità del metodo, sono stati replicati due esperimenti che hanno usato la procedura del condizionamento operante per indagare le capacità della gambusia (Gambusia holbrooki) di discriminare tra piccole numerosità e l’influenza del rapporto numerico e del numero totale di elementi nella discriminazione di grandi quantità (Agrillo et al., 2009, 2010). Nella nuova procedura, veniva introdotta una coppia di stimoli di diversa numerosità alle estremità della vasca sperimentale e successivamente veniva rilasciato del cibo in corrispondenza dello stimolo da rinforzare. I pesci sono stati inizialmente addestrati a distinguere un rapporto numerico relativamente semplice (0.5); successivamente nella fase di test, sono stati sottoposti a delle prove in estinzione (non veniva fornito il rinforzo alimentare) per verificare la loro capacità di generalizzare a nuove numerosità. La nuova procedura messa a punto ha replicato i risultati ottenuti con quella precedentemente utilizzata: i soggetti sono stati in grado di discriminare fino a 2 figure da 3; in presenza di grandi numerosità la prestazione diminuiva all’aumentare del rapporto numerico sebbene la loro capacità di discriminare sembri non avere un limite superiore. Il nuovo metodo si è inoltre rivelato rapido per la raccolta dei dati, applicabile a diverse specie di pesci ed efficacie per studiare l'apprendimento discriminativo in compiti che richiedono stimoli visivi. Di conseguenza, il nuovo protocollo è stato adottato in tutti gli esperimenti presentati in questa tesi che hanno usato la procedura di addestramento. Il secondo lavoro (Sezione 4.2) è incentrato su un potenziale limite della ricerca sulla cognizione numerica: la mancanza di studi inter-specifici che utilizzano la stessa metodologia. La questione se tutti i vertebrati condividano gli stessi sistemi numerici o se piuttosto le abilità numeriche siano apparse più volte durante l'evoluzione in risposta a specifiche pressioni selettive imposte dall'ambiente rappresenta uno dei temi principali della cognizione animale. Nonostante l’elevato numero di dati pubblicati, i risultati non sono coerenti dal momento che sono state utilizzate diverse metodologie di ricerca rendendo così difficile un confronto inter-specifico accurato. Ad oggi, nessuno studio ha indagato se diverse specie di pesci possiedano gli stessi sistemi numerici. Questo lavoro rappresenta il primo studio inter-specifico che utilizza la stessa metodologia nei pesci. Cinque diverse specie, la pecilia (Poecilia reticulata), lo zebrafish (Danio rerio), il pesce scalare (Pterophyllum scalare), la xenotoca (Xenotoca eiseni) ed il pesce combattente (Betta splendens), sono state inizialmente addestrate utilizzando un rapporto numerico semplice (0.50) e successivamente è stata confrontata la loro capacità di generalizzare a rapporti più difficili (0.67 e 0.75) o ad una numerosità maggiore (25 vs. 50) o minore (2 vs. 4). I risultati hanno mostrato interessanti somiglianze tra le specie, suggerendo la possibilità di sistemi numerici condivisi tra specie filogeneticamente distanti tra loro, più in accordo con l’esistenza di antichi sistemi di quantificazione ereditati da un antenato comune piuttosto che con un’evoluzione indipendente delle abilità numeriche in specie diverse. Un'altra questione importante nello studio della cognizione numerica riguarda l'influenza di fattori contestuali sulle capacità numeriche di una specie. È possibile che la prestazione osservata in un compito numerico sia limitata al contesto specifico in cui tali capacità sono state osservate piuttosto che riflettere le reali abilità numeriche della specie. Per questo motivo, il terzo (Sezione 5.1) e il quarto (Sezione 5.2) lavoro hanno studiato la potenziale influenza sull’accuratezza numerica dei pesci di fattori che normalmente si verificano in natura: il comportamento cooperativo all'interno del gruppo e la percezione di figure in movimento. In natura, gli animali che vivono in gruppo interagiscono tra di loro e queste interazioni ripetute tra gli individui possono incidere sulle scelte fatte in diversi contesti. Studi recenti hanno dimostrato che in alcune circostanze le azioni collettive permettono di aggirare i limiti cognitivi di una specie e di risolvere i problemi che vanno al di là delle capacità del singolo individuo (Krause et al., 2010, Couzin, 2009). Fino ad oggi, tutti gli studi di cognizione numerica condotti negli animali hanno preso in considerazione le prestazioni di singoli soggetti e non si sa quindi se il comportamento collettivo possa migliorare la capacità di risolvere compiti di discriminazione numerica. Lo scopo del terzo lavoro (Sezione 5.1) è stato quello di verificare se i pesci sottoposti a test in coppia fossero più accurati rispetto ai soggetti sottoposti a test individualmente in due diversi compiti di discriminazione numerica. Nel primo compito si è osservata la capacità delle pecilie di scegliere il gruppo di conspecifici più numeroso (4 vs. 6); nel secondo, invece, i pesci sono stati addestrati a discriminare tra gruppi di figure con un rapporto numerico pari a 0.5 e successivamente sono stati sottoposti a test usando confronti numerici più difficili (con rapporti pari a 0.67 e 0.75). I risultati hanno mostrato che i soggetti in coppia hanno avuto una prestazione migliore rispetto ai singoli, sia nella scelta del gruppo di conspecifici più numeroso, sia nel compito di discriminazione numerica, dimostrando quindi che il comportamento collettivo può fornire benefici che vanno al di là del singolo contesto ecologico. Inoltre, in entrambe le condizioni, il soggetto più accurato all’interno della coppia nella risoluzione del compito è emerso spontaneamente come leader. È interessante notare che i risultati ottenuti in questo lavoro sono in linea con i dati raccolti negli esseri umani adulti in cui la prestazione dei partecipanti in coppia è risultata superiore rispetto alle prestazioni individuali in un compito collettivo di discriminazione numerica (Bahrami et al., 2013). Questi dati suggeriscono quindi che la cooperazione aumenti l'acuità numerica in maniera simile in due specie filogeneticamente distanti tra di loro: gli esseri umani ed i pesci. Il movimento degli oggetti è un altro fattore che potrebbe potenzialmente influenzare l’acuità numerica. Gli animali sono infatti naturalmente esposti a degli elementi che si muovono (es. prede, predatori) e quindi il movimento rappresenta un segnale saliente nella loro vita. È stato dimostrato che la quantità di movimento dei conspecifici influenza in maniera differente la capacità dei pesci di discriminare tra piccoli (≤ 4) e grandi (≥ 4) gruppi di compagni sociali (Agrillo et al., 2008). Tuttavia non è stato ancora indagato se i pesci siano in grado di discriminare tra figure bidimensionali in movimento e se la loro accuratezza sia la stessa in presenza di piccole e grandi numerosità. Ad esempio, si è osservato che gli esseri umani adulti sono più veloci e più accurati nello stimare piccole quantità ( ≤ 4 ) di elementi in movimento piuttosto che grandi numerosità ( ≥ 4 ), supportando l'ipotesi di due sistemi numerici distinti (Trick et al., 2003, Alston & Humphreys, 2004). A tal fine, nel quarto lavoro (Sezione 5.2) esemplari di pecilia sono stati addestrati (con rapporto numerico 0.5) e sottoposti a test (con rapporto 0.67) con stimoli statici o in movimento. Si è osservato che gli elementi in movimento avevano un effetto simile a quello riportato nella nostra specie: mentre i soggetti a cui erano stati presentati gli stimoli statici non sono stati in grado di discriminate il rapporto pari a 0.67, sia in presenza di piccole che di grandi numerosità (3 vs. 4 e 9 vs. 12), i soggetti a cui erano stati presentati gli stimoli in movimento hanno saputo discriminare questo rapporto ma solo in presenza di piccole numerosità (3 vs 4). Ad oggi, nell’ambito della psicologia comparata c’è un dibattito sul fatto che gli animali posseggano un unico sistema di discriminazione per tutta la scala numerica (chiamato “Approximate number system”), oppure posseggano anche un sistema distinto coinvolto solo nella discriminazione di piccole numerosità (≤ 4) (chiamato “Object tracking system”). Sebbene i risultati ottenuti non rappresentino una prova diretta dell'esistenza di un sistema separato per la discriminazione numerica nell'intervallo 1-4 , il fatto che il movimento influenzi in maniera differente la discriminazione di piccole e grandi quantità nelle pecilie rafforza l'idea di sistemi cognitivi separati per piccoli e grandi numeri, in linea con i dati raccolti negli esseri umani. Nonostante in questa tesi non siano stati effettuati confronti diretti tra pesci e umani, è interessante notare le somiglianze osservate tra le due specie in quanto sollevano la possibilità che le nostre abilità numeriche abbiano un’origine più antica di quanto si pensi, che risalirebbe alla divergenza tra la linea evolutiva dei pesci e quella dei vertebrati terrestri . La seconda parte della tesi è incentrata sullo sviluppo delle abilità numeriche utilizzando gli avannotti di pecilia come specie modello. Gli studi sullo sviluppo delle abilità cognitive possono fornire indicazioni utili per quanto riguarda l'esistenza di un unico o più sistemi di rappresentazione numerica. Ad esempio, studiare lo sviluppo delle capacità numeriche in contesti diversi può aiutarci a capire se gli stessi sistemi numerici sono utilizzati in compiti diversi o piuttosto se vengono usati sistemi differenti. Dal momento che in letteratura non è presente un metodo adeguato per studiare l'apprendimento discriminativo in esemplari giovani di pecilia, nel quinto lavoro (Sezione 6.1) abbiamo messo a punto una procedura tenendo conto delle esigenze sociali dei giovani individui, al fine di ridurre al minimo il potenziale stress dovuto alla deprivazione sociale, senza interferire con il normale sviluppo del loro repertorio comportamentale. Pertanto, inizialmente abbiamo studiato lo sviluppo del comportamento sociale nelle prime due settimane di vita utilizzando un test di scelta spontanea dove gli avannotti potevano scegliere tra un compartimento contenente dei compagni sociali e uno compartimento vuoto. Successivamente veniva data ai giovani soggetti la possibilità di scegliere tra la propria immagine riflessa e un gruppo di coetanei per valutare se gli specchi potessero essere usati come sostituto dei compagni sociali durante gli esperimenti. Sulla base dei risultati ottenuti, è stato adattato il protocollo per l'apprendimento discriminativo usato nei pesci adulti per studiare la capacità degli avannotti di discriminare tra figure. I soggetti si sono dimostrati in grado di imparare una semplice discriminazione tra figure geometriche dopo poche prove; il metodo di addestramento è stato allora utilizzato nell’ultimo lavoro (Sezione 6.2) per studiare le loro capacità numeriche usando insiemi di figure bidimensionali, come viene comunemente fatto con i pesci adulti. Ad oggi, solamente Bisazza e collaboratori (2010) hanno studiato lo sviluppo ontogenetico delle abilità numeriche nei pesci. Gli autori hanno dimostrato che la capacità alla nascita delle pecilie di discriminare tra gruppi di conspecifici di diversa numerosità include tutti i confronti numerici nell’intervallo 1-4; i giovani soggetti hanno dimostrato inoltre di sapere discriminare piccole numerosità usando solamente l’informazione numerica. Nella Sezione 6.2 si è andato a verificare se gli avannotti di pecilia possono essere addestrati a discriminare tra insiemi di figure. A tal fine, sono state messe a punto tre condizioni sperimentali per studiare l'influenza delle variabili continue che co-variano con la numerosità (area complessiva degli stimoli, densità, ecc.). Nella prima condizione sia il numero che le variabili continue erano simultaneamente disponibili, nella seconda, solo l’informazione numerica era disponibile e, nell’ultima condizione l’informazione numerica è stata resa irrilevante (3 vs. 3) ed erano disponibili solo le variabili continue. Il risultato che i soggetti hanno saputo discriminare solo i confronti numerici facili nell’intervallo 1-4 quando sia il numero che le variabili continue erano disponibili è in contrasto con i dati ottenuti negli esperimenti di scelta spontanea (Bisazza et al., 2010), suggerendo che la capacità dei giovani pesci di utilizzare l’informazione numerica sia limitata agli stimoli sociali. Nel complesso i dati raccolti nella pecilia, sia in soggetti adulti che negli avannotti, suggeriscono l'esistenza nei pesci di molteplici meccanismi di discriminazione di quantità coinvolti nella risoluzione di problemi specifici, in accordo con l’ipotesi proposta in precedenza da diversi autori (Feigenson et al., 2004; Spelke, 2000). In sintesi, i dati raccolti in questa tesi indicano che anche i pesci, pur essendo dotati di un cervello molto più piccolo dei vertebrati a sangue caldo, possono discriminare tra quantità e risolvere compiti numerici complessi, in linea con altri ambiti di ricerca che suggeriscono come l’elaborazione dell’informazione numerica potrebbe non richiedere circuiti neurali complessi (Hope et al., 2010). Questo va di pari passo con la recente scoperta che i teleostei possiedono diverse abilità cognitive che in precedenza si ritenevano essere unicamente presenti nelle specie dotate di cervelli più grandi e complessi (es: mammiferi e specie di uccelli) (Bshary et al., 2002). Alla luce dei risultati presentati in questa tesi, è possibile affermare che i pesci costituiscono un modello adeguato per lo studio delle capacità cognitive ed in particolare di quelle numeriche.

In the last decades several studies have demonstrated that some numerical abilities are not strictly related to symbolic language and they are not a human prerogative. Recently the ability to discriminate between two quantities has been demonstrated also in fish. The aim of this project was to study the mechanisms underlying the non verbal numerical abilities in vertebrates. I investigated whether fish possess two independent numerical systems for the representation of small (<4) and large (>4) number or they possess a unique system that operates on the whole numerical scale. Both spontaneous choice and training experiments demonstrated that the ability to discriminate between large quantities was approximate and strongly dependent on the ratio between the numerosities, while discrimination in the small quantity range was not dependent on ratio and discriminating 3 from 4 was as easy as discriminating 1 from 4. The second part of the project regards the ontogeny of the ability to discriminate quantities. These experiments showed that the ability of fish to discriminate small numbers is innate and it is displayed immediately at birth while discrimination of large numbers emerges later as a result of both maturation and social experience. The third part concerns the role of non numerical variables in the discrimination of quantities. These experiments showed that fish are able to discriminate quantities even after the access to non numerical cues was made difficult and that learning a discrimination by using only numerical information is not more difficult than learning it by using only the non numerical variables. Finally, to investigate whether fish and humans share the same non verbal numerical systems, I carried out some experiments in which I compared fish and university students for their ability to discriminate the same numerical contrasts. Taken together these findings support the suggestion that discrete and continuous quantities are processed in humans and nonhuman animals by systems that evolved from a common ancestor more than 450 million years ago.
Negli ultimi anni è stato ampiamente dimostrato che le capacità numeriche non sono una prerogativa esclusivamente umana, infatti alcune abilità numeriche sono presenti anche in molte specie animali. Recentemente è stato dimostrato che anche i pesci sono in grado di compiere delle discriminazioni numeriche, e per questo sono stati utilizzati come modello sperimentale nel presente lavoro, allo scopo di approfondire lo studio dei meccanismi alla base delle abilità numeriche non verbali nei vertebrati. Una prima serie di esperimenti ha indagato se vi siano uno o due sistemi numerici non verbali nei pesci. I risultati, provenienti sia dagli esperimenti condotti con la procedura della scelta spontanea che da quelli condotti con la procedura di addestramento, hanno evidenziato come la capacità di discriminare grandi quantità (>4) sia fortemente influenzata dal rapporto numerico, mentre le discriminazioni tra piccole quantità (<4) non lo siano, in quanto discriminare 1 vs. 4 sarebbe facile quanto discriminare 3 vs. 4. Un secondo aspetto analizzato riguarda l’ontogenesi delle abilità numeriche. Questi esperimenti hanno evidenziato che nei pesci la capacità di discriminare piccole quantità è innata e presente fin dalla nascita, mentre quella di discriminare grandi quantità emerge più tardi come risultato della maturazione e dell’esperienza sociale. In seguito sono stati condotti una serie di esperimenti allo scopo di valutare il ruolo delle variabili non numeriche nelle discriminazioni di quantità: è emerso che i pesci sono in grado di discriminare diverse quantità anche quando l’accesso alle variabili non numeriche viene limitato; inoltre è stato dimostrato come per i pesci non sia più difficile apprendere una discriminazione sulla base della sola informazione numerica piuttosto che affidandosi alle sole variabili non numeriche, suggerendo che anche per i pesci il numero potrebbe essere una caratteristica primaria così come lo sono altre dimensioni degli stimoli. Infine in una serie di esperimenti le prestazioni dei pesci sono state confrontate con quelle degli esseri umani adulti in compiti paragonabili, allo scopo di verificare se i meccanismi alla base della discriminazione non verbale di quantità siano gli stessi in tutti i vertebrati. Nel loro insieme questi dati supportano l’ipotesi che sia nell’uomo che nei pesci le quantità siano elaborate attraverso dei sistemi che potrebbero essere evoluti da un comune antenato più di 450 milioni di anni fa.

Let B(H) denote the algebra of bounded linear operators on a complex Hilbert space H. The (classical) numerical range of T ∈ B(H) is the set W(T) = {〈T x; x〉: x ∈ H; ‖x‖ = 1} Writing T= T_1 + iT_2 for self-adjoint T_1, T_2 ∈ B(H), W(T) can be identified with the set {(〈T_1 x, x〉,〈T_2 x, x〉) : x ∈ H, ‖x‖ = 1}. This leads to the notion of the joint numerical range of T= 〖(T〗_1, T_2, …, T_n) ∈ 〖B(H)〗^n. It is defined by W(T) = {(〈T_1 x, x〉,〈T_2 x, x〉, …, 〈T_n x, x〉) : x ∈ H, ‖x‖ = 1}. The joint numerical range has been studied extensively in order to understand the joint behaviour of operators. Let K(H) be the set of all compact operators on a Hilbert space H. The essential numerical range of T ∈ B(H) is defined by W_(e ) (T)=∩{W(T+K) :K∈K(H) }. The joint essential numerical range of T= 〖(T〗_1, T_2, …, T_n) ∈〖 B(H)〗^n is defined analogously by W_(e ) (T)=∩{ /W(T+K) :K∈〖K(H)〗^n }. These notions have been generalized to operators on a Banach space. In Chapter 1 of this thesis, the joint spatial essential numerical range were introduced. Also the notions of the joint algebraic numerical range V(T) and the joint algebraic essential numerical range Ve(T) were reviewed. Basic properties of these sets were given. In 2010, Müller proved that each n-tuple of operators T on a separable Hilbert space has a compact perturbation T + K so that We(T) = W(T + K). In Chapter 2, it was shown that any n-tuple T of operators on lp has a compact perturbation T +K so that Ve(T) = V (T +K), provided that Ve(T) has an interior point. A key step was to find for each n-tuple of operators on lp a compact perturbation and a sequence of finite-dimensional subspaces with respect to which it is block 3 diagonal. This idea was inspired by a similar construction of Chui, Legg, Smith and Ward in 1979. Let H and L be separable Hilbert spaces and consider the operator D_AB=A⨂I_L⨂B on the tensor product space H ⨂▒L. In 1987 Magajna proved that W_(e ) (D_AB )=co[W_(e ) (A)- /(W(B)))∪/W(A) - W_(e ) (B))] by considering quasidiagonal operators. An alternative proof of the equality was given in Chapter 3 using block 3 diagonal operators. The maximal numerical range and the essential maximal numerical range of T ∈ B(H) were introduced by Stampi in 1970 and Fong in 1979 respectively. In 1993, Khan extended the notions to the joint essential maximal numerical range. However the set may be empty for some T ∈ B(H). In Chapter 4, the kth joint essential maximal numerical range, spatial maximal numerical range and algebraic numerical range were introduced. It was shown that kth joint essential maximal numerical range is non-empty and convex. Also, it was shown that the kth joint algebraic maximal numerical range is the convex hull of the kth joint spatial maximal numerical range. This extends the corresponding result of Fong.
published_or_final_version
Mathematics
Master
Master of Philosophy

Le présent travail se concentre sur les propriétés statistiques des scalaires réactifs subissant des réactions chimiques réversibles en turbulence incompressible. Une analyse théorique des propriétés statistiques des scalaires à différents ordres de moments a été réalisée sur la base d'approximations et de modèles convenablement proposés. Les résultats théoriquement dérivés ont ensuite été comparés aux résultats numériques obtenus par simulation numérique directe (DNS). Dans la simulation numérique directe, les dérivés spatiales ont été principalement approximées en utilisant une méthode pseudi-spectrale, car la vitesse turbulente et les champs scalaires sont généralement des conditions aux limites périodiques. Pour les configurations spéciales dans lesquelles la condition aux limites n'est pas périodique, une méthode aux différences finies avec des schémas fins a été utilisée pour approximer les dérivées spatiales. L'intégration temporelle numérique a été mise en oeuvre par un schéma Runge-Kutta du troisième ordre. Tous les travaux menés dans cette thèse sont consacrés aux explorations numériques et théoriques des scalaires réactifs en turbulence incompressible de différentes configurations. Nos résultats suggèrent de nouvelles idées pour de futures études, qui sont discutées dans les conclusions
The present work focuses on the statistical properties of reactive scalars undergoing reversible chemical reactions in incompressible turbulence. Theoretical analysis about the statistical properties of scalars at different order of moments were carried out based on appropriately proposed approximations and models. The theoretically derived results were then compared with numerical results obtained by direct numerical simulation (DNS). In the direct numerical simulation, the spatial derivatives were mainly approximated by using a pseudo-spectral method, since the turbulent velocity and scalar fields are generally of periodic boundary conditions. For the special configurations in which the boundary condition is not periodic, a finite difference method with fine schemes was used to approximate the spatial derivatives. The numerical time integration was implemented by a third order Runge-Kutta scheme. All the works carried out in this thesis are devoted to the numerical and theoretical explorations about reactive scalars is incompressible turbulence of different configurations. Our finding suggest new ideas for future studies, which are discussed in the conclusions

We consider computational methods for simulating the dynamics of molecular systems governed by the time-dependent Schrödinger equation. Solving the Schrödinger equation numerically poses a challenge due to its often highly oscillatory solutions, and to the exponential growth of work and memory with the number of particles in the system. Two different classes of problems are studied: the dynamics of the nuclei in a molecule and the dynamics of an electron in orbit around a nucleus. For the first class of problems we present new computational methods which exploit the relation between quantum and classical dynamics in order to make the computations more efficient. For the second class of problems, the lack of regularity in the solution poses a computational challenge. Using knowledge of the non-smooth features of the solution we construct a new method with two orders higher accuracy than what is achieved by direct application of a difference stencil.
eSSENCE

There are large individual differences in children s mathematical abilities when starting formal schooling and these differences can have lasting consequences. One factor that could lead to differences in children s mathematics skills is the home numeracy environment. This thesis examines the home numeracy environment, firstly as a whole concept and then more in-depth of one area of the home numeracy environment, number books. The home numeracy environment section starts by presenting a systematic review of the home numeracy environment literature and draws conclusions about the inconsistency of the results. The studies presented in this section investigate both methodological and theoretical questions surrounding the home numeracy environment. A novel text message method to measure the home numeracy environment is presented and the relationship between three different measures of the home numeracy environment (questionnaire, observation and text messages) is investigated, as well as their relationships to mathematics skills. This section has two key findings: firstly the self-report measures of the home numeracy environment are not related to the observation measure and secondly all three measures (apart from child number talk in the observation) were not related to mathematics skills. The second section of this thesis focuses on number books. Number books are often used in the home to teach young children number symbols. They primarily use multiple concrete pictures, but the benefits (or costs) to using these types of images are not known. The next three studies investigate the use of abstract and concrete images to teach children number symbols using an artificial symbol learning paradigm. It is concluded that there is a cost to using multiple representations when teaching children number symbols, and therefore number books should use a single picture throughout for children to benefit the most from the book. Overall the findings from this thesis show that the home numeracy environment is very broad and future research should change the way the home numeracy environment is measured and conduct more in-depth analysis of areas of the home numeracy environment.

A growing amount of evidence supports the hypothesis that humans are able, from the earliest age, to process numerical information in the absence of language. This work addresses the question of the nature of the internal representation for processing numerosities from three perspective: developmental, adults' skilled performance, and the peculiar case of synaesthesia. In our studies with children we addressed the development of the mental representation for numbers. In the first experiment we showed that, before formal teaching, preschoolers possess multiple numerical representations that follow a specific developmental trend. Indeed, they first rely on an intuitive representation where numbers are distributed logarithmically and progressively, with numerical practice and increasing knowledge, they shift to a formal and linear representation. Moreover, preschool children can exhibit both types of representations according to the familiarity with the context. In the second study, we tested the hypothesis that non-numerical sequences may also rely on a similar representation and follow the same developmental pattern. By studying children from the last year of kindergarten to 3rd grade we observed that numerical and non-numerical sequences have different mental representations. Indeed, only the numerical sequence shows the classical effects that support the hypothesis of a logarithmic representation. Moreover, we observed that children start to learn linearity in the numerical domain and then generalize the principle to all ordinal sequences. In our third study we investigated adults numerical representation of symbolic and non symbolic material. The aim was to test if the basic ability of discriminating between numerosities could explain higher level processes such as approximate calculation and symbolic number comparison. Indeed, if the preverbal approximate system of the numerical representation forms the basis of more complex numerical and mathematical knowledge, it should influence performance in other numerical tasks. Moreover, the crossing of symbolic and non-symbolic format of the stimuli for the approximate calculation task allowed us to qualify previous findings about the operational momentum effect in approximate arithmetic (i.e., the tendency to overestimate additions and underestimate subtractions). Indeed, we observed that the effect may be explained by the tendency to underestimate numerosities and that this bias is proportional to the set size. In the last experiment we investigated the relation between colour and numerical representation in NM, a number-colour synaesthete. Results showed that, in spite of not reporting colours for numerosities, our synaesthete was subject to interference effects. From these results we suggest a new model that accounts for the implicit and explicit synaesthetic effects by suggesting the existence of primary and secondary synaesthetic connections ("pseudo-synaesthesia"). Our results and model questions previous work on bi-directional effects and the operational definition of synaesthesia.

Lo scavo di gallerie rappresenta sicuramente una tra le sfide più impegnative che un ingegnere civile possa affrontare. Ciò è dovuto principalmente alla natura tridimensionale di questo problema di interazione terreno-struttura ma anche alle numerose incertezze che possono entrare in gioco nella progettazione. Recentemente, le tecniche di calcolo numeriche, che permettono una più ampia comprensione del problema, hanno subito un notevole sviluppo, diventando una risorsa fondamentale per la progettazione di scavi in sotterraneo. Tuttavia, solo ingegneri con una buona preparazione numerica sono in grado di gestire la modellazione di problemi di interazione terreno-struttura così complessi. Inoltre, tali modelli richiedono una attenta calibrazione dei parametri e una costante validazione con dati di monitoraggio. Lo scopo di questa tesi è quello di analizzare alcune delle principali problematiche legate alla progettazione di gallerie superficiali scavate in tradizionale. Il vantaggio principale dello scavo in traditionale rispetto a quello meccanizzato è legato alla maggiore flessibilità nella scelta dei rivestimenti e delle techniche di rinforzo del cavo e del fronte della galleria. Tuttavia, una maggiore flessibilità progettuale è necessariamente legata ad una profonda conoscenza del comportamento deformativo dell’ammasso, nonché ad un utilizzo consapevole delle tecniche modellazione numerica. Il presente lavoro è principalmente incentrato sulle seguenti tematiche riguardanti la progettazione di gallerie superficiali: - la stabilità di fronti di scavo rinforzati e non rinforzati; - l’applicabilità degli Eurocodici ad una progettazione condotta mediante tecniche di modellazione numerica; - la calibrazione dei parametri del modello numerico e la sua validazione attraverso dati di monitoraggio.
Among the problems that civil engineers have to face, the design and verification of an underground construction is one of the most challenging. A tunnel engineer has to tackle with a complex three-dimensional soil-structure interaction problem where many factors and uncertainties come into play. This is the reason why professional experience and engineering judgment usually play a crucial role. In recent years, numerical calculation techniques, which can provide an important basis for a better understanding of the problem, have strongly improved. They have become a fundamental resource for underground construction design, but they also entail some drawbacks: - only engineers with a strong numerical background can handle complex soil-structure interaction problems; - numerical calculations, especially if 3D, can be very time-consuming; - material parameters should be carefully evaluated, according to the particular problem and adopted constitutive law; - numerical models need to be validated with field monitoring data. The goal of this thesis is to investigate the main issues regarding the applicability of numerical analyses to the design and verification of traditionally excavated shallow tunnels. Despite, the remarkable technological improvement in mechanised tunnelling, traditional techniques still represent, in some cases, the most suitable and convenient solution. The principal advantage of traditional techniques is the high flexibility in the choice of supports and reinforcement measures. However, design flexibility implies a deep understanding of the ground response to underground openings as well as a conscious use of numerical models. This work provides a contribution to the numerical design of shallow tunnels by focusing on three principal issues: - stability of reinforced and unreinforced excavation faces; - Eurocodes applicability to a numerically-based design; - parameters calibration and numerical validation through comparison with monitoring data.

Multitud de problemas en ciencia e ingeniería se plantean como ecuaciones en derivadas parciales (EDPs). Si la frontera del recinto donde esas ecuaciones han de satisfacerse se desconoce a priori, se habla de "Problemas de frontera libre", propios de sistemas estacionarios no dependientes del tiempo, o bien de "Problemas de frontera móvil", asociados a problemas de evolución temporal, donde la frontera cambia con el tiempo. La solución a dichos problemas viene dada por la expresión de la(s) variable(s) dependiente(s) de la(s) EDP(s) junto con la función que determina la posición de la frontera. Dado que este tipo de problemas carece en la mayoría de los casos de solución analítica conocida, se hace preciso recurrir a métodos numéricos que permitan obtener una solución lo suficientemente aproximada, y que además mantenga propiedades cualitativas de la solución del modelo continuo de EDP(s). En este trabajo se ha abordado el estudio numérico de algunos problemas de frontera móvil provenientes de diversas disciplinas. La metodología aplicada consta de dos pasos sucesivos: aplicación de la transformación de Landau o "Front-fixing transformation" al modelo en EDP(s) con el fin de mantener inmóvil la frontera del dominio, y posterior discretización a través de un esquema en diferencias finitas. De ahí se obtienen esquemas numéricos que se implementan por medio de la herramienta MATLAB. Mediante un exhaustivo análisis numérico, se estudian propiedades del esquema y de la solución numérica (positividad, estabilidad, consistencia, monotonía, etc.). En el primer capítulo de este trabajo se revisa el estado del arte del campo objeto de estudio, se justifica la necesidad de disponer de métodos numéricos adaptados a este tipo de problemas y se describe brevemente la metodología empleada en nuestro enfoque. El Capítulo 2 se dedica a un problema perteneciente a la Biología Matemática y que consiste en determinar la evolución de la población de una especie invasora que se propaga en un hábitat. Este modelo consiste en una ecuación de difusión-reacción unida a una condición tipo Stefan. Los resultados del análisis numérico confirman la existencia de una dicotomía propagación-extinción en la evolución a largo plazo de la densidad de población de la especie invasora. En particular, se ha podido precisar el valor del coeficiente de la condición de Stefan que separa el comportamiento de propagación del de extinción. Los Capítulos 3 y 4 se centran en un problema de Química del Hormigón con interés en Ingeniería Civil: el proceso de carbonatación del hormigón, fenómeno evolutivo que lleva consigo la degradación progresiva de la estructura afectada y finalmente su ruina, si no se toman medidas preventivas. En el Capítulo 3 se considera un sistema de dos EDPs de tipo parabólico con dos incógnitas. Para su resolución, hay que considerar además las condiciones iniciales, las de contorno y las de tipo Stefan en la frontera. Los resultados numéricos confirman la tendencia de la ley de evolución de la frontera móvil hacia una función del tipo "raíz cuadrada del tiempo". En el Capítulo 4 se considera un modelo más general que el anterior, en el que intervienen seis especies químicas que se encuentran tanto en la zona carbonatada como en la no carbonatada. En el Capítulo 5 se aborda un problema de transmisión de calor que aparece en diversos procesos industriales; en este caso, en el enfriamiento durante la colada de metal fundido, donde la fase sólida avanza y la líquida se va extinguiendo. La frontera móvil (frente de solidificación) separa ambas fases, siendo su posición en cada instante la variable a determinar, junto con las temperaturas en cada fase. Después de la adecuada transformación y discretización, se implementa un esquema en diferencias finitas, subdividiendo el proceso en tres estadios temporales, a fin de tratar las singularidades asociadas a posicione
Many problems in science and engineering are formulated as partial differential equations (PDEs). If the boundary of the domain where these equations are to be solved is not known a priori, we face "Free-boundary problems", which are characteristic of non-time dependent stationary systems; besides, we have "Moving-boundary problems" in temporal evolution processes, where the border changes over time. The solution to these problems is given by the expression of the dependent variable(s) of PDE(s), together with the function that determines the position of the boundary. Since the analytical solution of this type of problems is lacked in most cases, it is necessary to resort to numerical methods that allow an accurate enough solution to be obtained, and which also maintain the qualitative properties of the solution(s) of the continuous model. This work approaches the numerical study of some moving-boundary problems that arise in different disciplines. The applied methodology consists of two successive steps: firstly, the so-called Landau transformation, or "Front-fixing transformation", which is used in the PDE(s) model to maintain the boundary of the domain immobile; later, we proceed to its discretization with a finite difference scheme. Different numerical schemes are obtained and implemented through the MATLAB computational tool. Properties of the scheme and the numerical solution (positivity, stability, consistency, monotonicity, etc.) are studied by an exhaustive numerical analysis. The first chapter of this work reports the state of the art of the field under study, justifies the need to adapt numerical methods to this type of problem, and briefly describes the methodology used in our approach. Chapter 2 presents a problem in Mathematical Biology that consists in determining over time the evolution of an invasive species population that spreads in a habitat. This problem is modelled by a diffusion-reaction equation linked to a Stefan-type condition. The results of the numerical analysis confirm the existence of a spreading-vanishing dichotomy in the long-term evolution of the population density of the invasive species. In particular, it is possible to determine the value of the coefficient of the Stefan condition that separates the propagation behaviour from extinction. Chapters 3 and 4 focus on a problem of Concrete Chemistry with an interest in Civil Engineering: the carbonation of concrete, an evolutionary phenomenon that leads to the progressive degradation of the affected structure and its eventual ruin if preventive measures are not taken. Chapter 3 considers a system of two parabolic type PDEs with two unknowns. For its resolution, the initial and boundary conditions have to be considered together with the Stefan conditions on the carbonation front. The numerical analysis results agree with those obtained in a previous theoretical study. The dynamics of the concentrations and the moving boundary confirm the long-term behaviour of the evolution law for the moving boundary as a "square root of time". Chapter 4 considers a more general model than the previous one, which includes six chemical species, defined in both the carbonated and non-carbonated zones, whose concentrations have to be found. Chapter 5 addresses a heat transfer problem that appears in various industrial processes; in this case, the solidification of metals in casting processes, where the solid phase advances and liquid reduces until it is depleted. The moving boundary (the solidification front) separates both phases. Its position in each instant is the variable to be determined together with the temperature profiles in both phases. After suitable transformation, discretization is carried out to obtain a finite difference scheme to be implemented. The process was subdivided into three temporal stages to deal with the singularities associated with the moving boundary position in the initialisation and depletion stages.
Multitud de problemes en ciència i enginyeria es plantegen com a equacions en derivades parcials (EDPs). Si la frontera del recinte on eixes equacions han de satisfer-se es desconeix a priori, es parla de "Problemas de frontera lliure", propis de sistemes estacionaris no dependents del temps, o bé de "Problemas de frontera mòbil", associats a problemes d'evolució temporal, on la frontera canvia amb el temps. Atés que este tipus de problemes manca en la majoria dels casos de solució analítica coneguda, es fa precís recórrer a mètodes numèrics que permeten obtindre una solució prou aproximada a l'exacta, i que a més mantinga propietats qualitatives de la solució del model continu d'EDP(s). En aquest treball s'ha abordat l'estudi numèric d'alguns problemes de frontera mòbil provinents de diverses disciplines. La metodologia aplicada consta de dos passos successius: en primer lloc, s'aplica l'anomenada transformació de Landau o "Front-fixing transformation" al model en EDP(s) a fi de mantindre immòbil la frontera del domini; posteriorment, es procedix a la seva discretització a través d'un esquema en diferències finites. D'ací s'obtenen esquemes numèrics que s'implementen per mitjà de la ferramenta informàtica MATLAB. Per mitjà d'una exhaustiva anàlisi numèrica, s'estudien propietats de l'esquema i de la solució numèrica (positivitat, estabilitat, consistència, monotonia, etc.). En el primer capítol d'aquest treball es revisa l'estat de l'art del camp objecte d'estudi, es justifica la necessitat de disposar de mètodes numèrics adaptats a aquest tipus de problemes i es descriu breument la metodologia emprada en el nostre enfocament. El Capítol 2 es dedica a un problema pertanyent a la Biologia Matemàtica i que consistix a determinar l'evolució en el temps de la distribució de la població d'una espècie invasora que es propaga en un hàbitat. Este model consistix en una equació de difusió-reacció unida a una condició tipus Stefan, que relaciona les funcions solució i frontera mòbil a determinar. Els resultats de l'anàlisi numèrica confirmen l'existència d'una dicotomia propagació-extinció en l'evolució a llarg termini de la densitat de població de l'espècie invasora. En particular, s'ha pogut precisar el valor del coeficient de la condició de Stefan que separa el comportament de propagació del d'extinció. Els Capítols 3 i 4 se centren en un problema de Química del Formigó amb interés en Enginyeria Civil: el procés de carbonatació del formigó, fenomen evolutiu que comporta la degradació progressiva de l'estructura afectada i finalment la seua ruïna, si no es prenen mesures preventives. En el Capítol 3 es considera un sistema de dos EDPs de tipus parabòlic amb dos incògnites. Per a la seua resolució, cal considerar a més, les condicions inicials, les de contorn i les de tipus Stefan en la frontera. Els resultats de l'anàlisi numèrica s'ajusten als obtinguts en un estudi teòric previ. S'han dut a terme experiments numèrics, comprovant la tendència de la llei d'evolució de la frontera mòbil cap a una funció del tipus "arrel quadrada del temps". En el Capítol 4 es considera un model més general, en el que intervenen sis espècies químiques les concentracions de les quals cal trobar, i que es troben tant en la zona carbonatada com en la no carbonatada. En el Capítol 5 s'aborda un problema de transmissió de calor que apareix en diversos processos industrials; en aquest cas, en el refredament durant la bugada de metall fos, on la fase sòlida avança i la líquida es va extingint. La frontera mòbil (front de solidificació) separa ambdues fases, sent la seua posició en cada instant la variable a determinar, junt amb les temperatures en cada una de les dos fases. Després de l'adequada transformació i discretització, s'implementa un esquema en diferències finites, subdividint el procés en tres estadis temporals, per tal de tractar les singularitats asso
Piqueras García, MÁ. (2018). Numerical Methods for Multidisciplinary Free Boundary Problems: Numerical Analysis and Computing [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/107948
TESIS

2013 - 2014
The subject of this thesis is the analysis and development of new numerical methods for Ordinary Di erential Equations (ODEs). This studies are motivated by the fundamental role that ODEs play in applied mathematics and applied sciences in general. In particular, as is well known, ODEs are successfully used to describe phenomena evolving in time, but it is often very di cult or even impossible to nd a solution in closed form, since a general formula for the exact solution has never been found, apart from special cases. The most important cases in the applications are systems of ODEs, whose exact solution is even harder to nd; then the role played by numerical integrators for ODEs is fundamental to many applied scientists. It is probably impossible to count all the scienti c papers that made use of numerical integrators during the last century and this is enough to recognize the importance of them in the progress of modern science. Moreover, in modern research, models keep getting more complicated, in order to catch more and more peculiarities of the physical systems they describe, thus it is crucial to keep improving numerical integrator's e ciency and accuracy. The rst, simpler and most famous numerical integrator was introduced by Euler in 1768 and it is nowadays still used very often in many situations, especially in educational settings because of its immediacy, but also in the practical integration of simple and well-behaved systems of ODEs. Since that time, many mathematicians and applied scientists devoted their time to the research of new and more e cient methods (in terms of accuracy and computational cost). The development of numerical integrators followed both the scienti c interests and the technological progress of the ages during whom they were developed. In XIX century, when most of the calculations were executed by hand or at most with mechanical calculators, Adams and Bashfort introduced the rst linear multistep methods (1855) and the rst Runge- Kutta methods appeared (1895-1905) due to the early works of Carl Runge and Martin Kutta. Both multistep and Runge-Kutta methods generated an incredible amount of research and of great results, providing a great understanding of them and making them very reliable in the numerical integration of a large number of practical problems. It was only with the advent of the rst electronic computers that the computational cost started to be a less crucial problem and the research e orts started to move towards the development of problem-oriented methods. It is probably possible to say that the rst class of problems that needed an ad-hoc numerical treatment was that of sti problems. These problems require highly stable numerical integrators (see Section ??) or, in the worst cases, a reformulation of the problem itself. Crucial contributions to the theory of numerical integrators for ODEs were given in the XX century by J.C. Butcher, who developed a theory of order for Runge-Kutta methods based on rooted trees and introduced the family of General Linear Methods together with K. Burrage, that uni ed all the known families of methods for rst order ODEs under a single formulation. General Linear Methods are multistagemultivalue methods that combine the characteristics of Runge-Kutta and Linear Multistep integrators... [edited by Author]
XIII n.s.

The main objective of this thesis is to provide a comprehensive numerical tool for the three-dimensional simulation of sedimentary basins. We have used a volume averaging technique to obtain a couple of basin-scale mathematical models. We have used some innovative numerical techniques to deal with such models. A multi-fluid implicit tracking technique is developed and integrated with a Stokes solver that is robust with respect to the variations of the coefficients. The movement of the basin boundaries and the evolution of the faults are treated with an Ale and a Finite Volume scheme respectively. Also some mesh refinement methods are used to guarantee a sufficient accuracy. The numerical experiments show a good qualitative agreement with the measured geometry of the sedimentary layers. (Pubblicata - vedi http://hdl.handle.net/2434/148441)

Finite element solution of solidification process in 2-D Cartesian and axisymmetric geometries is investigated. The use of finite element may result in spurious increase of temperature in the field and the selection of the mushy zone range when used as a numerical tool along with the selection of the mesh size results in large errors in the predicted solidification time. The approach works best for problems where the mushy zone range is finite and the thermal conductivities of both phases are high.

This thesis studies strongly coupled phases of condensed matter physics using a combination of gauge-gravity correspondence and numerical methods. We examine holographic models of the condensed matter phenomena of: vortex formation in the spontaneously broken phase of gauge theories, spontaneous breaking of translational invariance by periodic modulation, properties of (non-)Fermi liquids, and metal-insulator transitions in systems with sourced periodic modulation. In Chapter 2, we formulate a criterion for the existence of a Higgs phase based on the existence of bulk solitons. This criteria is applicable when the microscopic details of the gauge theory are unknown. We demonstrate the existence of such solitons in both top-down and bottom-up examples of holographic theories and examine their thermodynamics. In Chapter 3, we construct inhomogeneous, asymptotically Anti-deSitter Space (ADS) black hole solutions in Einstein-Maxwell-axion theory corresponding to the spontaneous breaking of translational invariance and the formation of striped order in the dual 2 + 1 dimensional Quantum Field Theory (QFT). We investigate the phase structure as function of parameters. In Chapter 4, we continue the study begun in Chapter 3. On domains of both fixed and variable wavenumber, we find a second order phase transition to the striped solution in each of the grand canonical, canonical and microcanonical ensembles. We also examine the properties of the bulk black hole solutions. In Chapter 5, we consider a phenomenological model whose bosonic sector is governed by the DBI action, and whose charged sector is purely fermionic. In this model, we demonstrate the existence of a compact worldvolume embedding, stabilized by a Fermi surface on a D-brane. We study the bulk and dual QFT thermodynamic and transport properties. In Chapter 6, we analyze low energy thermo-electric transport in a class of bottom-up, holographic models in which translation invariance has been broken. As a function of our choice of couplings, which parameterize this class of theories, we obtain (i) coherent metallic, or (ii) insulating, or (iii) incoherent metallic phases. We use a combination of analytical and numerical techniques to study the Alternating Current (AC) and Direct Current (DC) transport properties of these phases.
Science, Faculty of
Physics and Astronomy, Department of
Graduate

In the present study, a finite volume method is employed to modelthe advection-diffusion phenomenon during a pure substance meltingprocess. The exercise is limited to a benchmark problem consisting ofthe 2D melting from a vertical wall of a PCM driven by natural convectionin the melt. Numerical results, mainly the temporal evolutionof average Nusselt number at the hot wall and the average liquid fraction,are validated by available literature data and the effect of thermalinertia in the heat transfer is considered as well. Finally, motivatedby recent publications and the model presented here, possible new researchtopics are proposed.

This thesis presents the analysis, design and development of a multi-axis machinery numerical control system. The purpose of this research is to provide a numerical control method to overcome the multi-axis numerical control problem. The control system includes an IBM-PC computer as a host processor, a plug in multimotor controller board based on commercial numerical motor controller ICs. These implement a software specified digital control algorithm and output a PWM control number. Six motors, their driven actuators, and digital incremental feedback encoders complete the system. The experimental aspects of the work included the design of the IBM-PC plug in motor controller board and the motor drive board, computer software for real-time control and the system testing. For the theoretical aspects of the work, control theory was used to develop the mathematical model of the system. This aimed at providing a tool to predict and optimize the system performance, therefore, to fulfil the high positioning and high precision control task.

In the final book of the Georgics, Virgil digresses into a nostalgic and regretful explanation of his inability to include a proper discussion of gardening because he is spatiis exclusus iniquis (147). Often deemed “the skeleton of a fifth book of the Georgics” the exact meaning and intent behind this passage is still largely contested. In this paper I will attempt to de-strange this passage by examining it philosophically and allegorically, particularly by means of numerical symbolism.

The objective of this licentiate thesis was to study the effect of tool wear for sheet metal forming tools and how the wear process can be simulated in an efficient manner. Three Papers are appended to this licentiate thesis. Paper A covers the influence of tool geometry in deep drawing. In paper B is the way of calculating with finite element analysis described. The wear of a steel cylinder oscillating against a steel plate was studied experimentally. The worn shape of the cylinder was then compared with a numerical simulation of the shape. Paper C shows how numerical simulations can be used to simulate wear of deep drawing tools. The wear of two different deep drawing tools has been investigated. The shape of the tools before and after wear have been compared as well as the stresses and strains in the formed cups.
Godkänd; 2000; 20070317 (ysko)

The application of the finite integral of multiple variable functions is penetrating into more and more industries and science disciplines. The demands placed on solutions to these problems (such as high accuracy or high speed) are often quite contradictory. Therefore, it is not always possible to apply analytical approaches to these problems; numerical methods provide a suitable alternative. However, the ever-growing complexity of these problems places too high a demand on many of these numerical methods, and so neither of these methods are useful for solving such problems. The goal of this thesis is to design and implement a new numerical method that provides highly accurate and very fast computation of finite integrals of multiple variable functions. This new method combines pre-existing approaches in the field of numerical mathematics.

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This paper examines the production of numerical relations done by students of a 4th grade of elementary school at a public school in the city of Sao Paulo, S.P., Brazil. Several studies, in particular the ones done by Gimenez & Lins, Kamii and Franchi, show the need of establishing relations between the numbers, identifying meaning for the numbers and operations as a flexible way to solve problems. This flexibility can be searched through the interaction between arithmetic and geometric domains. Therefore, a series of activities were applied in order to search this flexibility. At first, these activities mobilized counting processes, notion of unity, quantitative relations interacted with geometry, particularly through the use of notions of perimeter and area as tools, according to the elements of didactics, tool-object and the interaction of domains developed by Douady. A confrontation was provoked between the notions of linear and bilinear magnitude through changes occurred on the sides of the rectangle, on its perimeter and area. These changes, according to Rogalski, have a deep relation with the addition and multiplicative structures. The use of the graph paper tries to favour not only the visual perception of the unity and the display of these unities in rectangular arrangement but also the comprehension of the area calculus and the multiplicative procedures. Both the records and the analysis of the data allowed concluding that the students initially set up quantitative relations such as the part-whole , single and multiple by establishing meaning for the numerical relations in the determination of numerical expressions. Thus, composition and decomposition of rectangular shape activities occurred in the relation part-whole , single and the multiple not only in the formation of new unities but also in the numerical relations. The findings of this study provided evidence that with the production of numerical relations, the students gave sense for the expressions, showed self-confidence and flexibility in the answers given
Este trabalho estuda a produção de relações numéricas por alunos da quarta série do ensino fundamental, em uma escola pública do município de São Paulo. Diversos trabalhos, como os de Gimenez & Lins, Kamii e Franchi revelam a necessidade de se estabelecer relações entre os números, de se identificar significado para os números e operações como uma forma flexível de resolver problemas. Essa flexibilidade pode ser buscada por meio da interação entre domínios aritméticos e geométricos. Para tanto, aplicamos uma série de atividades que inicialmente mobilizou processos de contagem, noção de unidade, relações quantitativas, interadas pela geometria, particularmente pelo uso das noções de perímetro e área como ferramentas, segundo os elementos de didática, ferramenta-objeto e interação de domínios, desenvolvidos por Douady. Provocamos um confronto entre a noção de grandezas lineares e bilineares por variações ocorridas nos lados do retângulo, no seu perímetro e na sua área as quais segundo Rogalski possuem profunda relação com as estruturas aditivas e multiplicativas. A utilização de papel quadriculado busca favorecer a percepção visual da unidade e a disposição dessas unidades em arranjo retangular e ainda favorecer a compreensão do cálculo de área e dos procedimentos multiplicativos. A análise dos dados e registros tomados permitiu concluir que os alunos estabeleceram inicialmente relações quantitativas, como a de parte-todo, uno e múltiplo estabelecendo sentido para as relações numéricas na formação de expressões aritméticas. Assim, atividades de composição e decomposição de figuras retangulares incidiram sobre a relação parte-todo, uno e o múltiplo tanto na formação de novas unidades como na formação de relações numéricas. Os resultados obtidos mostraram que com a produção de relações numéricas os alunos deram sentido para as expressões, apresentaram autoconfiança e flexibilidade nas respostas apresentadas

Direct numerical simulation (DNS) is used to study some aspects of the dynamics of vortex rings in viscous, incompressible ow at Reynolds numbers (defined as the ratio of the initial circulation to the kinematic viscosity) in the range of 103 to 104. Firstly, the effect of the particular initial core azimuthal vorticity profile of a vortex ring on its subsequent evolution in unbounded ow is studied. Vortex rings with a wide range of initial core vorticity profiles are shown to relax to a common equilibrium state. Additionally the behaviour of the equilibrium vortex ring at large times is studied. When the slenderness ratio of the vortex rings increases beyond a particular limit, the vortex rings diverge from the common equilibrium state and follow paths determined by the viscosity of the uid. Secondly, the interaction of a laminar vortex ring with a non-deformable, free-slip surface at an oblique angle of incidence leading to the phenomenon of vortex reconnection is investigated. Specifically the effect of Reynolds number on the dynamics of the reconnection process is studied. The scaling of the reconnection timescale with the Reynolds number is obtained. At high Reynolds numbers the reconnection process leads to a breakdown of the entire vortex ring structure to a turbulent-like ow. This phenomenon is shown to be related to the mechanics of the reconnection process. Finally, the dynamics of vortex rings with swirl in unbounded ow is studied. Two different types of vortex rings with swirl were considered: i) Vortex rings with Gaussian distributions of core azimuthal vorticity and core azimuthal velocity and ii) Steady state solutions of the Euler equations for vortex rings with swirl. Both types of vortex rings develop an elongated axial vortex after initialisation. The existence of a maximum limit for the swirl on a vortex ring is shown above which the vortex rings undergo a rapid de-swirling readjustment. A helical instability occurring in vortex rings due to swirl at high Reynolds numbers is presented. A relation is shown to exist between one of the modes of the helical instability and the geometric parameters of the vortex ring.

The aim of this thesis work is the study of the turbulent entrainment phenomenon in jets through numerical experiments. More specifically, an attempt to study the effect of engulfment and nibbling mechanisms separately was made. The flow chosen for the numerical experiments is the temporal planar jet. The idea behind these experiments is to study the spreading and the mixing of a passive scalar under the effect of two modified velocity fields. The first is a large-scale velocity field obtained through a filtering operation, while the second is a small-scale velocity field obtained subtracting the large-scale velocity field from the total one and then adding the mean velocity. Initially, the post-processing of a spatially developing planar jet, performed by Doctor Andrea Fregni and Professor Andrea Cimarelli, has been carried out in order to analyse the main features of spatially evolving jets compared with the temporal ones. A co-flow and a passive scalar are present in the simulation. The Reynolds number is set to Re = 3000 and the Schmidt number is Sc = 1. After this first step, a benchmark DNS of a temporal planar jet with Re = 3000 and Sc = 1 has been performed in order to evaluate the main differences with respect to the spatially evolving jet. Once the settings were validated, the numerical experiments with large and small scale velocity fields have been performed. The filter used in all the experiments is the box filter. The results of two different filter lengths are presented, the first is Δ = 1.5λcl and the second is Δ = 3λcl. Since λcl is function of time, the two filter lengths are themselves varying in time. The results of the experiments were then compared with those of the unfiltered solution. The passive scalar spread approximatively the same amount under the effect of the large-scale velocity fields and under the effect of the unfiltered velocity. On the other hand, the small-scale fluctuations have been proved to be important in the mixing process.

We study the numerical integration of nonholonomic problems. The problems are formulated using Lagrangian and Hamiltonian mechanics. We review briefly the theoretical concepts used in geometric mechanics. We reconstruct two nonholonomic variational integrators from the monograph of Monforte. We also construct two one-step integrators based on a combination of the continuous Legendre transform and the discrete Legendre transform from an article by Marsden and West. Inintially these integrators display promising behavior, but they turn out to be unstable. The variational integrators are compared with a classical Runge-Kutta method. We compare the methods on three nonholonomic systems: The nonholonomic particle from the monograph of Monforte, the nonholonomic system of particles from an article by McLachlan and Perlmutter, and a variation of the Chaplygin sleigh from Bloch.

We discuss nonholonomic systems in general and numerical methods for solving them. Two different approaches for obtaining numerical methods are considered; discretization of the Lagrange-d'Alembert equations on the one hand, and using the discrete Lagrange-d'Alembert principle to obtain nonholonomic integrators on the other. Among methods using the first approach, we focus on the super partitioned additive Runge-Kutta (SPARK) methods. Among nonholonomic integrators, we focus on a reversible second order method by McLachlan and Perlmutter. Through several numerical experiments the methods we present are compared by considering error-growth, conservation of energy, geometric properties of the solution and how well the constraints are satisfied. Of special interest is the comparison of the 2-stage SPARK Lobatto IIIA-B method and the nonholonomic integrator by McLachlan and Perlmutter, which both are reversible and of second order. We observe a clear connection between energy-conservation and the geometric properties of the numerical solution. To preserve energy in long-time integrations is seen to be important in order to get solutions with the correct qualitative properties. Our results indicate that the nonholonomic integrator by McLachlan and Perlmutter sometimes conserves energy better than the 2-stage SPARK Lobatto IIIA-B method. In a recent work by Jay, however, the same two methods are compared and are found to conserve energy equally well in long-time integrations.

One of the most conspicuous problems in space physics for the last decades has been to theoretically describe how the large parallel electric fields on auroral field lines can be generated. There is strong observational evidence of such electric fields, and stationary theory supports the need for electric fields accelerating electrons to the ionosphere where they generate auroras. However, dynamic models have not been able to reproduce these electric fields. This thesis sheds some light on this incompatibility and shows that the missing ingredient in previous dynamic models is a correct description of the electron temperature. As the electrons accelerate towards the ionosphere, their velocity along the magnetic field line will increase. In the converging magnetic field lines, the mirror force will convert much of the parallel velocity into perpendicular velocity. The result of the acceleration and mirroring will be a velocity distribution with a significantly higher temperature in the auroral acceleration region than above. The enhanced temperature corresponds to strong electron pressure gradients that balance the parallel electric fields. Thus, in regions with electron acceleration along converging magnetic field lines, the electron temperature increase is a fundamental process and must be included in any model that aims to describe the build up of parallel electric fields. The development of such a model has been hampered by the difficulty to describe the temperature variation. This thesis shows that a local equation of state cannot be used, but the electron temperature variations must be descibed as a nonlocal response to the state of the auroral flux tube. The nonlocal response can be accomplished by the particle-fluid model presented in this thesis. This new dynamic model is a combination of a fluid model and a Particle-In-Cell (PIC) model and results in large parallel electric fields consistent with in-situ observations.

The effects of the temperature and length, of the preheat zone, on the deflagration to detonation transition are investigated through numerical simulation. The Navier-Stokes equations, with a reaction term, are solved in one dimension. The time integration is a one-dimensional adaptation of an existing two-dimensional finite volume method code. An iterative scheme, based on an overlap integral, is developed for the determination of the deflagration to detonation transition. The code is tested in a number of cases, where the analytical solution (to the Euler equations) is known. The location of the deflagration to detonation transition is displayed graphically through the preheat zone temperature as a function of the fuel mixture temperature, for fixed exhaust gas temperature and with the preheat zone length as a parameter. The evolution of the deflagration to detonation transition is investigated for an initial state well within the regime where the deflagration to detonation transition occurs. Graphs displaying the temporal evolution of pressure, temperature, reaction rate, and fuel mass fraction are presented. Finally, a method for estimating the flame velocity during the deflagration and detonation phases, as well as the flame acceleration during the intermediate phase, is developed.

Predictive accuracy of the previously developed coupled code for the solution of the time-dependent Navier-Stokes equations in conjunction with the radiative transfer equation was first assessed by applying it to the prediction of thermally radiating, hydrodynamically developed laminar pipe flow for which the numerical solution had been reported in the literature. The effect of radiation on flow and temperature fields was demonstrated for different values of conduction to radiation ratio. It was found that the steady-state temperature predictions of the code agree well with the benchmark solution. In an attempt to test the predictive accuracy of the coupled code for turbulent radiating flows, it was applied to fully developed turbulent flow of a hot gas through a relatively cold pipe and the results were compared with the numerical solution available in the literature. The code was found to mimic the reported steady-state temperature profiles well. Having validated the predictive accuracy of the coupled code for steady, laminar/turbulent, radiating pipe flows, the performance of the code for transient radiating flows was tested by applying it to a test problem involving laminar/turbulent flow of carbon dioxide through a circular pipe for the simulation of simultaneous hydrodynamic and thermal development. The transient solutions for temperature, velocity and radiative energy source term fields were found to demonstrate the physically expected trends. In order to improve the performance of the code, a parallel algorithm of the code was developed and tested against sequential code for speed up and efficiency. It was found that the same results are obtained with a reasonably high speed-up and efficiency.

Conformal map has application in a lot of areas of science, e.g., fluid flow, heat conduction, solidification, electromagnetic, etc. Especially conformal map applied to elasticity theory can provide most simple and useful solution. But finding of conformal map for custom domain is not trivial problem. We used a numerical method for building a conformal map to solve torsion problem. In addition it was considered an infinite system method to solve the same problem. Results are compared.

A general introduction to conformal maps and the Riemann mapping theorem is given. Three methods for numerically approximating conformal maps from arbitrary domains to the unit disc are presented: the Schwarz-Christoffel method, the geodesic algorithm and the circle packing method. Basic implementations of the geodesic algorithm and the circle packing method were made, and program code is presented. Applications of these numerical methods to problems in physics and mathematical research are briefly discussed.

The main objective of this thesis is to describe the transient thermodynamics during physisorption of hydrogen gas using a commercial numerical software.Simulations of thermal effects during adsorption are valuable tools for the efficient design of hydrogen adsorption storage systems. Transient mass and energy equations are used for describing the adsorption process. For this purpose, experimental adsorption data has to be presented analytically. Several models have been developed for this objective.The thesis consists of two parts. In the first, a literature study on adsorption theories and thermodynamic assumptions for development of transient mass and energy balances is conducted. The models are discussed, and from this, the Langmuir approach is selected to be used for numerical calculations. The model is implemented into a lumped-parameter analysis describing an infinitesimal element within an adsorbent bed, allowing for neglecting heat leaks into the system as well as the structural steel mass. The second part describes the simulations conducted in the study. The numerical software COMSOL Multiphysics 4.2.a is used for numerical calculations. Modules for implementation of the transient mass and energy balances are considered, before Heat Transfer in Porous Media and Brinkman Equations are applied, for heat transfer, pressure- and velocity calculations, respectively. The simulations are run for different initial and boundary conditions. The porous material is defined with Fe-btc properties. The simulation model is built step by step, and problems encountered are analyzed continuously in the process towards a complete model. After completion, the model geometry is adjusted and the porous material is changed to MOF-5 properties, to resemble a selected published paper. Numerical results are compared and discussed. Modeling restrictions for the present study is accounted for, and all choices made when considering the assigned task are justified. The report is completed by listing the conclusions drawn from the present study, and concrete suggestions for further work are given.Simulation results found in the present study differs slightly from the published research work. Instabilities in the solver results in a temperature dip in the simulated domain. This leads to an increased adsorption rate. Furthermore, it appears that mass is not conserved, which means that the inlet velocity of the feed gas does not change as expected when the adsorption is disabled from the model.

This thesis is on the topic of numerical thermal analysis, specically of the SmallExplorer for Advanced Missions SEAM. SEAM is a 3 unit Cubesat, which isgoing to be launched in a sun-synchronous orbit to measure the magnetic sphere.It makes use of a boom deployment system to remove the sensors from themagnetic eld inuences of the body. The goal of this thesis is to study thethermal behaviour of the satellite, specically the internal components and thethermal deformation of the boom structure. The numerical simulations makeuse of the Monte Carlo Ray-tracing method. Furthermore thermal vacuumcycle tests have been compared to the thermal model as a form of validation.Additionally the thesis also serves as a nal thermal analysis of the satellite, tocheck if all components operate within their specied thermal operating range.
Detta examensarbete handlar om numerisk termisk analys av SEAM (SmallExplorer for Advanced Missions) satellit. SEAM är en 3U CubeSat, som skaskickas upp i solsynkron bana kring jorden för att utföra magnetfältmätningar.Satelliten använder sig av en utfällbar bom för att separera magnetsensorer frånmagnetiska störningar från satellitens elektronik. Examensarbetet syftar tillatt studera termiska beteende av satelliten, specifikt temperaturområden i bananför interna komponenter samt termisk deformation av den utfällbara bomstrukturen.Numeriska simuleringar av strålningsöverföring av värme använderMonte-Carlo metod för att följa strålar. Experimentella resultat från termiskvakuum testning av satelliten har jämförts med termiska modellen för att valideraden. Examensarbetet utgör den slutliga termiska analysen av satelliten, föratt säkerställa att alla komponenter används inom deras specificerade temperaturområde.