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Hardy inequalities on graphs
(2024)
The dissertation deals with a central inequality of non-linear potential theory, the Hardy inequality. It states that the non-linear energy functional can be estimated from below by a pth power of a weighted p-norm, p>1. The energy functional consists of a divergence part and an arbitrary potential part. Locally summable infinite graphs were chosen as the underlying space. Previous publications on Hardy inequalities on graphs have mainly considered the special case p=2, or locally finite graphs without a potential part.
Two fundamental questions now arise quite naturally: For which graphs is there a Hardy inequality at all? And, if it exists, is there a way to obtain an optimal weight? Answers to these questions are given in Theorem 10.1 and Theorem 12.1. Theorem 10.1 gives a number of characterizations; among others, there is a Hardy inequality on a graph if and only if there is a Green's function. Theorem 12.1 gives an explicit formula to compute optimal Hardy weights for locally finite graphs under some additional technical assumptions. Examples show that Green's functions are good candidates to be used in the formula.
Emphasis is also placed on illustrating the theory with examples. The focus is on natural numbers, Euclidean lattices, trees and star graphs. Finally, a non-linear version of the Heisenberg uncertainty principle and a Rellich inequality are derived from the Hardy inequality.
Resolving the evolutionary history of two hippotragin antelopes using archival and ancient DNA
(2024)
African antelopes are iconic but surprisingly understudied in terms of their genetics, especially when it comes to their evolutionary history and genetic diversity. The age of genomics provides an opportunity to investigate evolution using whole nuclear genomes. Decreasing sequencing costs enable the recovery of multiple loci per genome, giving more power to single specimen analyses and providing higher resolution insights into species and populations that can help guide conservation efforts. This age of genomics has only recently begun for African antelopes. Many African bovids have a declining population trend and hence, are often endangered. Consequently, contemporary samples from the wild are often hard to collect. In these cases, ex situ samples from contemporary captive populations or in the form of archival or ancient DNA (aDNA) from historical museum or archaeological/paleontological specimens present a great research opportunity with the latter two even offering a window to information about the past. However, the recovery of aDNA is still considered challenging from regions with prevailing climatic conditions that are deemed adverse for DNA preservation like the African continent. This raises the question if DNA recovery from fossils as old as the early Holocene from these regions is possible.
This thesis focuses on investigating the evolutionary history and genetic diversity of two species: the addax (Addax nasomaculatus) and the blue antelope (Hippotragus leucophaeus). The addax is critically endangered and might even already be extinct in the wild, while the blue antelope became extinct ~1800 AD, becoming the first extinct large African mammal species in historical times. Together, the addax and the blue antelope can inform us about current and past extinction events and the knowledge gained can help guide conservation efforts of threatened species. The three studies used ex situ samples and present the first nuclear whole genome data for both species. The addax study used historical museum specimens and a contemporary sample from a captive population. The two studies on the blue antelope used mainly historical museum specimens but also fossils, and resulted in the recovery of the oldest paleogenome from Africa at that time.
The aim of the first study was to assess the genetic diversity and the evolutionary history of the addax. It found that the historical wild addax population showed only limited phylogeographic structuring, indicating that the addax was a highly mobile and panmictic population and suggesting that the current European captive population might be missing the majority of the historical mitochondrial diversity. It also found the nuclear and mitochondrial diversity in the addax to be rather low compared to other wild ungulate species. Suggestions on how to best save the remaining genetic diversity are presented. The European zoo population was shown to exhibit no or only minor levels of inbreeding, indicating good prospects for the restoration of the species in the wild. The trajectory of the addax’s effective population size indicated a major bottleneck in the late Pleistocene and a low effective population size well before recent human impact led to the species being critically endangered today.
The second study set out to investigate the identities of historical blue antelope specimens using aDNA techniques. Results showed that six out of ten investigated specimens were misidentified, demonstrating the blue antelope to be one of the scarcest mammal species in historical natural history collections, with almost no bone reference material. The preliminary analysis of the mitochondrial genomes suggested a low diversity and hence low population size at the time of the European colonization of southern Africa.
Study three presents the results of the analyses of two blue antelope nuclear genomes, one ~200 years old and another dating to the early Holocene, 9,800–9,300 cal years BP. A fossil-calibrated phylogeny dated the divergence time of the three historically extant Hippotragus species to ~2.86 Ma and demonstrated the blue and the sable antelope (H. niger) to be sister species. In addition, ancient gene flow from the roan (H. equinus) into the blue antelope was detected. A comparison with the roan and the sable antelope indicated that the blue antelope had a much lower nuclear diversity, suggesting a low population size since at least the early Holocene. This concurs with findings from the fossil record that show a considerable decline in abundance after the Pleistocene–Holocene transition. Moreover, it suggests that the blue antelope persisted throughout the Holocene regardless of a low population size, indicating that human impact in the colonial era was a major factor in the blue antelope’s extinction.
This thesis uses aDNA analyses to provide deeper insights into the evolutionary history and genetic diversity of the addax and the blue antelope. Human impact likely was the main driver of extinction in the blue antelope, and is likely the main factor threatening the addax today. This thesis demonstrates the value of ex situ samples for science and conservation, and suggests to include genetic data for conservation assessments of species. It further demonstrates the beneficial use of aDNA for the taxonomic identification of historically important specimens in natural history collections. Finally, the successful retrieval of a paleogenome from the early Holocene of Africa using shotgun sequencing shows that DNA retrieval from samples of that age is possible from regions generally deemed unfavorable for DNA preservation, opening up new research opportunities. All three studies enhance our knowledge of African antelopes, contributing to the general understanding of African large mammal evolution and to the conservation of these and similarly threatened species.
Background: Societies worldwide have become more diverse yet continue to be inequitable. Understanding how youth growing up in these societies are socialized and consequently develop racial knowledge has important implications not only for their well-being but also for building more just societies. Importantly, there is a lack of research on these topics in Germany and Europe in general.
Aim and Method: The overarching aim of the dissertation is to investigate 1) where and how ethnic-racial socialization (ERS) happens in inequitable societies and 2) how it relates to youth’s development of racial knowledge, which comprises racial beliefs (e.g., prejudice, attitudes), behaviors (e.g., actions preserving or disrupting inequities), and identities (e.g., inclusive, cultural). Guided by developmental, cultural, and ecological theories of socialization and development, I first explored how family, as a crucial socialization context, contributes to the preservation or disruption of racism and xenophobia in inequitable societies through its influence on children’s racial beliefs and behaviors. I conducted a literature review and developed a conceptual model bridging research on ethnic-racial socialization and intergroup relations (Study 1). After documenting the lack of research on socialization and development of racial knowledge within and beyond family contexts outside of the U.S., I conducted a qualitative study to explore ERS in Germany through the lens of racially marginalized youth (Study 2). Then, I conducted two quantitative studies to explore the separate and interacting relations of multiple (i.e., family, school) socialization contexts for the development of racial beliefs and behaviors (Study 3), and identities (Studies 3, 4) in Germany. Participants of Study 2 were 26 young adults (aged between 19 and 32) of Turkish, Kurdish, East, and Southeast Asian heritage living across different cities in Germany. Study 3 was conducted with 503 eighth graders of immigrant and non-immigrant descent (Mage = 13.67) in Berlin, Study 4 included 311 early to mid-adolescents of immigrant descent (Mage= 13.85) in North Rhine-Westphalia with diverse cultural backgrounds.
Results and Conclusion: The findings revealed that privileged or marginalized positions of families in relation to their ethnic-racial and religious background in society entail differential experiences and thus are an important determining factor for the content/process of socialization and development of youth’s racial knowledge. Until recently, ERS research mostly focused on investigating how racially marginalized families have been the sources of support for their children in resisting racism and how racially privileged families contribute to transmission of information upholding racism (Study 1). ERS for racially marginalized youth in Germany centered heritage culture, discrimination, and resistance strategies to racism, yet resistance strategies transmitted to youth mostly help to survive racism (e.g., working hard) by upholding it instead of liberating themselves from racism by disrupting it (e.g., self-advocacy, Study 2). Furthermore, when families and schools foster heritage and intercultural learning, both contexts may separately promote stronger identification with heritage culture and German identities, and more prosocial intentions towards disadvantaged groups (i.e., refugees) among youth (Studies 3, 4). However, equal treatment in the school context led to mixed results: equal treatment was either unrelated to inclusive identity, or positively related to German and negatively related to heritage culture identities (Studies 3, 4). Additionally, youth receiving messages highlighting strained and preferential intergroup relations at home while attending schools promoting assimilation may develop a stronger heritage culture identity (Study 4). In conclusion, ERS happened across various social contexts (i.e., family, community centers, school, neighborhood, peer). ERS promoting heritage and intercultural learning, at least in one social context (family or school), might foster youth’s racial knowledge manifesting in stronger belonging to multiple cultures and in prosocial intentions toward disadvantaged groups. However, there is a need for ERS targeting increasing awareness of discrimination across social contexts of youth and teaching youth resistance strategies for liberation from racism.
During the last decades, therapeutical proteins have risen to great significance in the pharmaceutical industry. As non-human proteins that are introduced into the human body cause a distinct immune system reaction that triggers their rapid clearance, most newly approved protein pharmaceuticals are shielded by modification with synthetic polymers to significantly improve their blood circulation time. All such clinically approved protein-polymer conjugates contain polyethylene glycol (PEG) and its conjugation is denoted as PEGylation. However, many patients develop anti-PEG antibodies which cause a rapid clearance of PEGylated molecules upon repeated administration. Therefore, the search for alternative polymers that can replace PEG in therapeutic applications has become important. In addition, although the blood circulation time is significantly prolonged, the therapeutic activity of some conjugates is decreased compared to the unmodified protein. The reason is that these conjugates are formed by the traditional conjugation method that addresses the protein's lysine side chains. As proteins have many solvent exposed lysines, this results in a somewhat uncontrolled attachment of polymer chains, leading to a mixture of regioisomers, with some of them eventually affecting the therapeutic performance.
This thesis investigates a novel method for ligating macromolecules in a site-specific manner, using enzymatic catalysis. Sortase A is used as the enzyme: It is a well-studied transpeptidase which is able to catalyze the intermolecular ligation of two peptides. This process is commonly referred to as sortase-mediated ligation (SML). SML constitutes an equilibrium reaction, which limits product yield. Two previously reported methods to overcome this major limitation were tested with polymers without using an excessive amount of one reactant.
Specific C- or N-terminal peptide sequences (recognition sequence and nucleophile) as part of the protein are required for SML. The complementary peptide was located at the polymer chain end. Grafting-to was used to avoid damaging the protein during polymerization. To be able to investigate all possible combinations (protein-recognition sequence and nucleophile-protein as well as polymer-recognition sequence and nucleophile-polymer) all necessary building blocks were synthesized. Polymerization via reversible deactivation radical polymerization (RDRP) was used to achieve a narrow molecular weight distribution of the polymers, which is required for therapeutic use.
The synthesis of the polymeric building blocks was started by synthesizing the peptide via automated solid-phase peptide synthesis (SPPS) to avoid post-polymerization attachment and to enable easy adaptation of changes in the peptide sequence. To account for the different functionalities (free N- or C-terminus) required for SML, different linker molecules between resin and peptide were used.
To facilitate purification, the chain transfer agent (CTA) for reversible addition-fragmentation chain-transfer (RAFT) polymerization was coupled to the resin-immobilized recognition sequence peptide. The acrylamide and acrylate-based monomers used in this thesis were chosen for their potential to replace PEG.
Following that, surface-initiated (SI) ATRP and RAFT polymerization were attempted, but failed. As a result, the newly developed method of xanthate-supported photo-iniferter (XPI) RAFT polymerization in solution was used successfully to obtain a library of various peptide-polymer conjugates with different chain lengths and narrow molar mass distributions.
After peptide side chain deprotection, these constructs were used first to ligate two polymers via SML, which was successful but revealed a limit in polymer chain length (max. 100 repeat units). When utilizing equimolar amounts of reactants, the use of Ni2+ ions in combination with a histidine after the recognition sequence to remove the cleaved peptide from the equilibrium maximized product formation with conversions of up to 70 %.
Finally, a model protein and a nanobody with promising properties for therapeutical use were biotechnologically modified to contain the peptide sequences required for SML. Using the model protein for C- or N-terminal SML with various polymers did not result in protein-polymer conjugates. The reason is most likely the lack of accessibility of the protein termini to the enzyme. Using the nanobody for C-terminal SML, on the other hand, was successful. However, a similar polymer chain length limit was observed as in polymer-polymer SML. Furthermore, in case of the synthesis of protein-polymer conjugates, it was more effective to shift the SML equilibrium by using an excess of polymer than by employing the Ni2+ ion strategy.
Overall, the experimental data from this work provides a good foundation for future research in this promising field; however, more research is required to fully understand the potential and limitations of using SML for protein-polymer synthesis. In future, the method explored in this dissertation could prove to be a very versatile pathway to obtain therapeutic protein-polymer conjugates that exhibit high activities and long blood circulation times.
Die Digitalisierung ist ein wesentlicher Bestandteil aktueller Verwaltungsreformen. Trotz der hohen Bedeutung und langjähriger Bemühungen bleibt die Bilanz der Verwaltungsdigitalisierung in Deutschland ambivalent. Diese Studie konzentriert sich auf drei erfolgreiche Digitalisierungsvorhaben aus dem Onlinezugangsgesetz (OZG) und analysiert mittels problemzentrierter Expertenbefragung Einflussfaktoren auf die Umsetzung von OZG-Vorhaben und den Einfluss des Managements in diesem Prozess. Die Analyse erfolgt theoriegeleitet basierend auf dem Ansatz der begrenzten Rationalität und der ökonomischen Theorie der Bürokratie. Die Ergebnisse zeigen, dass anzunehmen ist, dass die identifizierten Einflussfaktoren unterschiedlich auf Nachnutzbarkeit und Reifegrad von Verwaltungsleistungen wirken und als Folgen begrenzter Rationalität im menschlichen Problemlösungsprozess interpretiert werden können. Managerinnen unterstützen die operativen Akteure bei der Umsetzung, indem sie deren begrenzte Rationalität mit geeigneten Strategien adressieren. Dazu können sie Ressourcen bereitstellen, mit ihrer Expertise unterstützen, Informationen zugänglich machen, Entscheidungswege verändern sowie zur Konfliktlösung beitragen. Die Studie bietet wertvolle Einblicke in die tatsächliche Managementpraxis und leitet daraus Empfehlungen für die Umsetzung öffentlicher Digitalisierungsvorhaben sowie für die Steuerung öffentlicher Verwaltungen ab. Diese Studie liefert einen wichtigen Beitrag zum Verständnis des Einflusses des Managements in der Verwaltungsdigitalisierung. Die Studie unterstreicht außerdem die Notwendigkeit weiterer Forschung in diesem Bereich, um die Praktiken und Herausforderungen der Verwaltungsdigitalisierung besser zu verstehen und effektiv zu adressieren.
This thesis explores word order variability in verb-final languages. Verb-final languages have a reputation for a high amount of word order variability. However, that reputation amounts to an urban myth due to a lack of systematic investigation. This thesis provides such a systematic investigation by presenting original data from several verb-final languages with a focus on four Uralic ones: Estonian, Udmurt, Meadow Mari, and South Sámi. As with every urban myth, there is a kernel of truth in that many unrelated verb-final languages share a particular kind of word order variability, A-scrambling, in which the fronted elements do not receive a special information-structural role, such as topic or contrastive focus. That word order variability goes hand in hand with placing focussed phrases further to the right in the position directly in front of the verb. Variations on this pattern are exemplified by Uyghur, Standard Dargwa, Eastern Armenian, and three of the Uralic languages, Estonian, Udmurt, and Meadow Mari. So far for the kernel of truth, but the fourth Uralic language, South Sámi, is comparably rigid and does not feature this particular kind of word order variability. Further such comparably rigid, non-scrambling verb-final languages are Dutch, Afrikaans, Amharic, and Korean. In contrast to scrambling languages, non-scrambling languages feature obligatory subject movement, causing word order rigidity next to other typical EPP effects.
The EPP is a defining feature of South Sámi clause structure in general. South Sámi exhibits a one-of-a-kind alternation between SOV and SAuxOV order that is captured by the assumption of the EPP and obligatory movement of auxiliaries but not lexical verbs. Other languages that allow for SAuxOV order either lack an alternation because the auxiliary is obligatorily present (Macro-Sudan SAuxOVX languages), or feature an alternation between SVO and SAuxOV (Kru languages; V2 with underlying OV as a fringe case). In the SVO–SAuxOV languages, both auxiliaries and lexical verbs move. Hence, South Sámi shows that the textbook difference between the VO languages English and French, whether verb movement is restricted to auxiliaries, also extends to OV languages. SAuxOV languages are an outlier among OV languages in general but are united by the presence of the EPP.
Word order variability is not restricted to the preverbal field in verb-final languages, as most of them feature postverbal elements (PVE). PVE challenge the notion of verb-finality in a language. Strictly verb-final languages without any clause-internal PVE are rare. This thesis charts the first structural and descriptive typology of PVE. Verb-final languages vary in the categories they allow as PVE. Allowing for non-oblique PVE is a pivotal threshold: when non-oblique PVE are allowed, PVE can be used for information-structural effects. Many areally and genetically unrelated languages only allow for given PVE but differ in whether the PVE are contrastive. In those languages, verb-finality is not at stake since verb-medial orders are marked. In contrast, the Uralic languages Estonian and Udmurt allow for any PVE, including information focus. Verb-medial orders can be used in the same contexts as verb-final orders without semantic and pragmatic differences. As such, verb placement is subject to actual free variation. The underlying verb-finality of Estonian and Udmurt can only be inferred from a range of diagnostics indicating optional verb movement in both languages. In general, it is not possible to account for PVE with a uniform analysis: rightwards merge, leftward verb movement, and rightwards phrasal movement are required to capture the cross- and intralinguistic variation.
Knowing that a language is verb-final does not allow one to draw conclusions about word order variability in that language. There are patterns of homogeneity, such as the word order variability driven by directly preverbal focus and the givenness of postverbal elements, but those are not brought about by verb-finality alone. Preverbal word order variability is restricted by the more abstract property of obligatory subject movement, whereas the determinant of postverbal word order variability has to be determined in the future.
The automotive industry is a prime example of digital technologies reshaping mobility. Connected, autonomous, shared, and electric (CASE) trends lead to new emerging players that threaten existing industrial-aged companies. To respond, incumbents need to bridge the gap between contrasting product architecture and organizational principles in the physical and digital realms. Over-the-air (OTA) technology, that enables seamless software updates and on-demand feature additions for customers, is an example of CASE-driven digital product innovation. Through an extensive longitudinal case study of an OTA initiative by an industrial- aged automaker, this dissertation explores how incumbents accomplish digital product innovation. Building on modularity, liminality, and the mirroring hypothesis, it presents a process model that explains the triggers, mechanisms, and outcomes of this process. In contrast to the literature, the findings emphasize the primacy of addressing product architecture challenges over organizational ones and highlight the managerial implications for success.
Human activities modify nature worldwide via changes in the environment, biodiversity and the functioning of ecosystems, which in turn disrupt ecosystem services and feed back negatively on humans. A pressing challenge is thus to limit our impact on nature, and this requires detailed understanding of the interconnections between the environment, biodiversity and ecosystem functioning. These three components of ecosystems each include multiple dimensions, which interact with each other in different ways, but we lack a comprehensive picture of their interconnections and underlying mechanisms. Notably, diversity is often viewed as a single facet, namely species diversity, while many more facets exist at different levels of biological organisation (e.g. genetic, phenotypic, functional, multitrophic diversity), and multiple diversity facets together constitute the raw material for adaptation to environmental changes and shape ecosystem functioning. Consequently, investigating the multidimensionality of ecosystems, and in particular the links between multifaceted diversity, environmental changes and ecosystem functions, is crucial for ecological research, management and conservation. This thesis aims to explore several aspects of this question theoretically.
I investigate three broad topics in this thesis. First, I focus on how food webs with varying levels of functional diversity across three trophic levels buffer environmental changes, such as a sudden addition of nutrients or long-term changes (e.g. warming or eutrophication). I observed that functional diversity generally enhanced ecological stability (i.e. the buffering capacity of the food web) by increasing trophic coupling. More precisely, two aspects of ecological stability (resistance and resilience) increased even though a third aspect (the inverse of the time required for the system to reach its post-perturbation state) decreased with increasing functional diversity. Second, I explore how several diversity facets served as a raw material for different sources of adaptation and how these sources affected multiple ecosystem functions across two trophic levels. Considering several sources of adaptation enabled the interplay between ecological and evolutionary processes, which affected trophic coupling and thereby ecosystem functioning. Third, I reflect further on the multifaceted nature of diversity by developing an index K able to quantify the facet of functional diversity, which is itself multifaceted. K can provide a comprehensive picture of functional diversity and is a rather good predictor of ecosystem functioning. Finally I synthesise the interdependent mechanisms (complementarity and selection effects, trophic coupling and adaptation) underlying the relationships between multifaceted diversity, ecosystem functioning and the environment, and discuss the generalisation of my findings across ecosystems and further perspectives towards elaborating an operational biodiversity-ecosystem functioning framework for research and conservation.
Homomorphisms are a fundamental concept in mathematics expressing the similarity of structures. They provide a framework that captures many of the central problems of computer science with close ties to various other fields of science. Thus, many studies over the last four decades have been devoted to the algorithmic complexity of homomorphism problems. Despite their generality, it has been found that non-uniform homomorphism problems, where the target structure is fixed, frequently feature complexity dichotomies. Exploring the limits of these dichotomies represents the common goal of this line of research.
We investigate the problem of counting homomorphisms to a fixed structure over a finite field of prime order and its algorithmic complexity. Our emphasis is on graph homomorphisms and the resulting problem #_{p}Hom[H] for a graph H and a prime p. The main research question is how counting over a finite field of prime order affects the complexity.
In the first part of this thesis, we tackle the research question in its generality and develop a framework for studying the complexity of counting problems based on category theory. In the absence of problem-specific details, results in the language of category theory provide a clear picture of the properties needed and highlight common ground between different branches of science. The proposed problem #Mor^{C}[B] of counting the number of morphisms to a fixed object B of C is abstract in nature and encompasses important problems like constraint satisfaction problems, which serve as a leading example for all our results. We find explanations and generalizations for a plethora of results in counting complexity. Our main technical result is that specific matrices of morphism counts are non-singular. The strength of this result lies in its algebraic nature. First, our proofs rely on carefully constructed systems of linear equations, which we know to be uniquely solvable. Second, by exchanging the field that the matrix is defined by to a finite field of order p, we obtain analogous results for modular counting. For the latter, cancellations are implied by automorphisms of order p, but intriguingly we find that these present the only obstacle to translating our results from exact counting to modular counting. If we restrict our attention to reduced objects without automorphisms of order p, we obtain results analogue to those for exact counting. This is underscored by a confluent reduction that allows this restriction by constructing a reduced object for any given object. We emphasize the strength of the categorial perspective by applying the duality principle, which yields immediate consequences for the dual problem of counting the number of morphisms from a fixed object.
In the second part of this thesis, we focus on graphs and the problem #_{p}Hom[H]. We conjecture that automorphisms of order p capture all possible cancellations and that, for a reduced graph H, the problem #_{p}Hom[H] features the complexity dichotomy analogue to the one given for exact counting by Dyer and Greenhill. This serves as a generalization of the conjecture by Faben and Jerrum for the modulus 2. The criterion for tractability is that H is a collection of complete bipartite and reflexive complete graphs. From the findings of part one, we show that the conjectured dichotomy implies dichotomies for all quantum homomorphism problems, in particular counting vertex surjective homomorphisms and compactions modulo p. Since the tractable cases in the dichotomy are solved by trivial computations, the study of the intractable cases remains. As an initial problem in a series of reductions capable of implying hardness, we employ the problem of counting weighted independent sets in a bipartite graph modulo prime p. A dichotomy for this problem is shown, stating that the trivial cases occurring when a weight is congruent modulo p to 0 are the only tractable cases. We reduce the possible structure of H to the bipartite case by a reduction to the restricted homomorphism problem #_{p}Hom^{bip}[H] of counting modulo p the number of homomorphisms between bipartite graphs that maintain a given order of bipartition. This reduction does not have an impact on the accessibility of the technical results, thanks to the generality of the findings of part one. In order to prove the conjecture, it suffices to show that for a connected bipartite graph that is not complete, #_{p}Hom^{bip}[H] is #_{p}P-hard. Through a rigorous structural study of bipartite graphs, we establish this result for the rich class of bipartite graphs that are (K_{3,3}\{e}, domino)-free. This overcomes in particular the substantial hurdle imposed by squares, which leads us to explore the global structure of H and prove the existence of explicit structures that imply hardness.
Among the different meanings carried by numerical information, cardinality is fundamental for survival and for the development of basic as well as of higher numerical skills. Importantly, the human brain inherits from evolution a predisposition to map cardinality onto space, as revealed by the presence of spatial-numerical associations (SNAs) in humans and animals. Here, the mapping of cardinal information onto physical space is addressed as a hallmark signature characterizing numerical cognition.
According to traditional approaches, cognition is defined as complex forms of internal information processing, taking place in the brain (cognitive processor). On the contrary, embodied cognition approaches define cognition as functionally linked to perception and action, in the continuous interaction between a biological body and its physical and sociocultural environment.
Embracing the principles of the embodied cognition perspective, I conducted four novel studies designed to unveil how SNAs originate, develop, and adapt, depending on characteristics of the organism, the context, and their interaction. I structured my doctoral thesis in three levels. At the grounded level (Study 1), I unfold the biological foundations underlying the tendency to map cardinal information across space; at the embodied level (Study 2), I reveal the impact of atypical motor development on the construction of SNAs; at the situated level (Study 3), I document the joint influence of visuospatial attention and task properties on SNAs. Furthermore, I experimentally investigate the presence of associations between physical and numerical distance, another numerical property fundamental for the development of efficient mathematical minds (Study 4).
In Study 1, I present the Brain’s Asymmetric Frequency Tuning hypothesis that relies on hemispheric asymmetries for processing spatial frequencies, a low-level visual feature that the (in)vertebrate brain extracts from any visual scene to create a coherent percept of the world. Computational analyses of the power spectra of the original stimuli used to document the presence of SNAs in human newborns and animals, support the brain’s asymmetric frequency tuning as a theoretical account and as an evolutionarily inherited mechanism scaffolding the universal and innate tendency to represent cardinality across horizontal space.
In Study 2, I explore SNAs in children with rare genetic neuromuscular diseases: spinal muscular atrophy (SMA) and Duchenne muscular dystrophy (DMD). SMA children never accomplish independent motoric exploration of their environment; in contrast, DMD children do explore but later lose this ability. The different SNAs reported by the two groups support the critical role of early sensorimotor experiences in the spatial representation of cardinality.
In Study 3, I directly compare the effects of overt attentional orientation during explicit and implicit processing of numerical magnitude. First, the different effects of attentional orienting based on the type of assessment support different mechanisms underlying SNAs during explicit and implicit assessment of numerical magnitude. Secondly, the impact of vertical shifts of attention on the processing of numerical distance sheds light on the correspondence between numerical distance and peri-personal distance.
In Study 4, I document the presence of different SNAs, driven by numerical magnitude and numerical distance, by employing different response mappings (left vs. right and near vs. distant).
In the field of numerical cognition, the four studies included in the present thesis contribute to unveiling how the characteristics of the organism and the environment influence the emergence, the development, and the flexibility of our attitude to represent cardinal information across space, thus supporting the predictions of the embodied cognition approach. Furthermore, they inform a taxonomy of body-centred factors (biological properties of the brain and sensorimotor system) modulating the spatial representation of cardinality throughout the course of life, at the grounded, embodied, and situated levels.
If the awareness for different variables influencing SNAs over the course of life is important, it is equally important to consider the organism as a whole in its sensorimotor interaction with the world. Inspired by my doctoral research, here I propose a holistic perspective that considers the role of evolution, embodiment, and environment in the association of cardinal information with directional space. The new perspective advances the current approaches to SNAs, both at the conceptual and at the methodological levels.
Unveiling how the mental representation of cardinality emerges, develops, and adapts is necessary to shape efficient mathematical minds and achieve economic productivity, technological progress, and a higher quality of life.