@phdthesis{13331, abstract = {The extension of extremal combinatorics to the setting of exterior algebra is a work in progress that gained attention recently. In this thesis, we study the combinatorial structure of exterior algebra by introducing a dictionary that translates the notions from the set systems into the framework of exterior algebra. We show both generalizations of celebrated Erdös--Ko--Rado theorem and Hilton--Milner theorem to the setting of exterior algebra in the simplest non-trivial case of two-forms. }, author = {Köse, Seyda}, issn = {2791-4585}, pages = {26}, publisher = {Institute of Science and Technology Austria}, title = {{Exterior algebra and combinatorics}}, doi = {10.15479/at:ista:13331}, year = {2023}, } @phdthesis{14226, abstract = {We introduce the notion of a Faustian interchange in a 1-parameter family of smooth functions to generalize the medial axis to critical points of index larger than 0. We construct and implement a general purpose algorithm for approximating such generalized medial axes.}, author = {Stephenson, Elizabeth R}, issn = {2791-4585}, pages = {43}, publisher = {Institute of Science and Technology Austria}, title = {{Generalizing medial axes with homology switches}}, doi = {10.15479/at:ista:14226}, year = {2023}, } @phdthesis{12531, abstract = {All visual experiences of the vertebrates begin with light being converted into electrical signals by the eye retina. Retinal ganglion cells (RGCs) are the neurons of the innermost layer of the mammal retina, and they transmit visual information to the rest of the brain. It has been shown that RGCs vary in their morphology and genetic profiles, moreover they can be unambiguously grouped into subtypes that share the same morphological and/or molecular properties. However, in terms of RGCs function, it remains unclear how many distinct types there are and what response properties their typology relies on. Even given the recent studies that successfully classified RGCs in a patch of the retina [1] and in scotopic conditions [2], the question remains whether the found subtypes persist across the entire retina. In this work, using a novel imaging method, we show that, when sampled from a large portion of the retina, RGCs can not be clearly divided into functional subtypes. We found that in photopic conditions, which implies more prominent natural scene statistic differences across the visual field, response properties can be exhibited by cells differently depending on their location in the retina, which leads to formation of a gradient of features rather than distinct classes. This finding suggests that RGCs follow a global organization across the visual field of the animal, adapting each RGC subtype to the requirements imposed by the natural scene statistics.}, author = {Kirillova, Kseniia}, issn = {2791-4585}, pages = {46}, publisher = {Institute of Science and Technology Austria}, title = {{Panoramic functional gradients across the mouse retina}}, doi = {10.15479/at:ista:12531}, year = {2023}, } @phdthesis{12800, abstract = {The evolutionary processes that brought about today’s plethora of living species and the many billions more ancient ones all underlie biology. Evolutionary pathways are neither directed nor deterministic, but rather an interplay between selection, migration, mutation, genetic drift and other environmental factors. Hybrid zones, as natural crossing experiments, offer a great opportunity to use cline analysis to deduce different evolutionary processes - for example, selection strength. Theoretical cline models, largely assuming uniform distribution of individuals, often lack the capability of incorporating population structure. Since in reality organisms mostly live in patchy distributions and their dispersal is hardly ever Gaussian, it is necessary to unravel the effect of these different elements of population structure on cline parameters and shape. In this thesis, I develop a simulation inspired by the A. majus hybrid zone of a single selected locus under frequency dependent selection. This simulation enables us to untangle the effects of different elements of population structure as for example a low-density center and long-range dispersal. This thesis is therefore a first step towards theoretically untangling the effects of different elements of population structure on cline parameters and shape. }, author = {Julseth, Mara}, issn = {2791-4585}, pages = {21}, publisher = {Institute of Science and Technology Austria}, title = {{The effect of local population structure on genetic variation at selected loci in the A. majus hybrid zone}}, doi = {10.15479/at:ista:12800}, year = {2023}, } @phdthesis{10422, abstract = {Those who aim to devise new materials with desirable properties usually examine present methods first. However, they will find out that some approaches can exist only conceptually without high chances to become practically useful. It seems that a numerical technique called automatic differentiation together with increasing supply of computational accelerators will soon shift many methods of the material design from the category ”unimaginable” to the category ”expensive but possible”. Approach we suggest is not an exception. Our overall goal is to have an efficient and generalizable approach allowing to solve inverse design problems. In this thesis we scratch its surface. We consider jammed systems of identical particles. And ask ourselves how the shape of those particles (or the parameters codifying it) may affect mechanical properties of the system. An indispensable part of reaching the answer is an appropriate particle parametrization. We come up with a simple, yet generalizable and purposeful scheme for it. Using our generalizable shape parameterization, we simulate the formation of a solid composed of pentagonal-like particles and measure anisotropy in the resulting elastic response. Through automatic differentiation techniques, we directly connect the shape parameters with the elastic response. Interestingly, for our system we find that less isotropic particles lead to a more isotropic elastic response. Together with other results known about our method it seems that it can be successfully generalized for different inverse design problems.}, author = {Piankov, Anton}, issn = {2791-4585}, publisher = {Institute of Science and Technology Austria}, title = {{Towards designer materials using customizable particle shape}}, doi = {10.15479/at:ista:10422}, year = {2021}, }