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    Kaplansky's problem and unitary orbits in matrix amplifications
    (Theta Foundation, 2025) Marcoux, Laurent W.; Sarkowicz, Pawel; Zhang, Yuanhang
    We study the distances between the unitary orbits of matrix amplifications of elements in certain C*-algebras. In particular, we show that the distance between unitary orbits of arbitrary elements in unital, separale, UHF-stable C*-algebras remains unchanged when amplifying to certain matrix sizes. We further exhibit examples of elements in C*-algebras where the distance between unitary orbits becomes strictly smaller after amplifying by a certain matrix size, and we demonstrate that distances between unitary orbits of amplifications are not monotone in the multiplicity of the amplifications, even in the setting of matrix algebras. Lastly, we show that topological K-theory provides obstructions in the purely infinite setting.
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    Stability relations for Hilbert space operators and a problem of Kaplansky
    (Springer Nature, 2024) Marcoux, Laurent W.; Radjavi, Heydar; Troscheit, Sascha; Zhang, Yuanhang
    In his monograph on Infinite Abelian Groups, I. Kaplansky raised three "test problems" concerning their structure and multiplicity. As noted by Azoff, these problems make sense for any category admitting a direct sum operation. Here, we are interested in the operator theoretic version of Kaplansky's second problem which asks: if A and B are operators on an infinite-dimensional, separable Hilbert space and A ⊕ A is equivalent to B ⊕ B in some (precise) sense, is A equivalent to B? We examine this problem under a strengthening of the hypothesis, where a "primitive" square root J2(A) of A ⊕ A is assumed to be equivalent to the corresponding square root J2(B) of B ⊕ B. When "equivalence" refers to similarity of operators and A is a compact operator, we deduce from this stronger hypothesis that A and B are similar. We exhibit a counterexample (due to J. Bell) of this phenomenon in the setting of unital rings. Also, we exhibit an uncountable family {Uα}α∈ of unitary operators, no two of which are unitarily equivalent, such that each Uα is unitarily equivalent to Jn(Uα), a “primitive” nth root of Uα ⊕ Uα ⊕···⊕ Uα.
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    Carbon Cycling, Carbon Storage, and Greenhouse Gas Dynamics in Temperate Southern Ontario Swamps
    (University of Waterloo, 2026-11-10) Schmidt, Megan
    Swamps are treed wetlands and account for the highest areal extent of wetland types in Southern Ontario due to the conversion of other wetlands for development and agriculture. However, little is known about the environmental factors and processes that influence carbon (C) dynamics or the resulting C stocks and fluxes therein. Additionally, historically soils were considered the primary or only source of methane (CH4) efflux from swamps, but recent studies have shown that tree stems contribute a significant amount of the total ecosystem CH4 emissions. Swamps are relatively understudied in Canada, therefore uncertainties remain in the true extent of swamps nationally, that, combined with the lack of data, has limited their inclusion in C accounting, climate modelling, and policy. A multi-year study of temperate swamps was conducted across southern Ontario to assess the exchange of C, carbon dioxide (CO2) and CH4 between soil, trees, and atmosphere, and estimate current C stocks in soil and biomass. To do so, we began with a synoptic study of 12 swamps over 12 months that looked at the effects of hydrogeomorphic setting and canopy type on measurable variables of water table and soil temperature and the subsequent effects on CH4 and CO2. We found mean growing season soil respiration of CO2 and CH4 across the swamps ranged from 11–49 g m-2 d-1 and -3–44 mg m-2 d-1, respectively; biomass stored 7.92–37.0 kg m-2 of C across the sites, and soil stored 11.6–35.5 kg m-2. Over the study year, the swamps functioned as C sinks and followed similar patterns of relationships to environmental conditions as bogs and fens, with little difference between hydrogeomorphic settings or canopy types. To assess the contribution of tree stems to CH4 flux, three common genera (Acer, Betula, and Thuja) within 3 of the previous 12 swamps were measured simultaneously with soil fluxes over 21 months. As most CH4 emitted from tree stems in wetlands is produced within the soil and transported up the stem, we also measured soil CH4 fluxes alongside environmental conditions such as water table and soil temperature. We found that stem CH₄ emissions varied substantially among species, sites, and seasons, and were influenced by soil temperature, water-table position, and soil CH₄ availability. However, the importance of these drivers differed among sites and tree species, indicating that methane transport through trees is controlled by interacting environmental and physiological processes rather than a single universal mechanism. The highest mean fluxes were from Acer at 0.23 – 20.82 mg CH4 m-2 of stem yr-1, followed by Betula at 0.09 – 0.89 mg CH4 m-2 yr-1, and Thuja at 0.12 – 0.63 mg CH4 m-2 yr-1. Finally, these studies were combined to create estimates of the full current C stocks and fluxes for four of the swamps. We found that C storage was dominated by soils at 20 – 86 kg C m-2 of swamp, followed by living trees at 9.8 – 19.9 kg C m-2. Woody debris contained 2.9 – 12.5 kg C m-2, while the contribution from living understory vegetation was minimal at 0.1 kg C m-2 or less. Total C stored within the sites ranged from 45.6 to 108.3 kg C m-2. Trees and understory vegetation took up 619 – 982 g C m-2 of swamp yr-1 but were offset by soil and tree fluxes of 898 – 1213 g C m-2 of swamp yr-1, resulting in three swamps acting as C sources at 25 – 298 and one swamp acting as a C sink at -108 g C m-2 of swamp yr-1. In conclusion, temperate swamps in southern Ontario represent locally important stocks of both biomass and soil C, surpassing both upland forests and other wetland classes. Carbon dynamics in temperate swamps are driven by different controls across timescales, with long-term C storage determined by peat accumulation and climate, while short-term greenhouse gas fluxes are regulated by temperature-moisture interactions that respond to the changing seasons and annual weather. As a result, future warming may increase CH4 emissions, particularly if conditions are able to sustain high water tables that support anaerobic processes. Conversely, increased prolonged dry periods may enhance CO2 losses while limiting CH4 emissions, potentially shifting the balance of GHG outputs, but could be offset by enhanced overstory productivity and litter inputs. Fragmentation, conversion, and drainage result in isolated remnants, and these remnant swamps still account for a large portion of wetland in southern Ontario. Both within and surrounding swamps, anthropogenic disturbances may create drier conditions, in turn increasing soil C oxidation and reducing C stores and peat accumulation ability. Incorporating these data into C accounting and modelling will help inform future policies of wetland conservation and restoration and management of swamps as natural climate solutions.
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    Around the closures of the set of commutators and the set of differences of idempotent elements of B(H)
    (Theta Foundation, 2024) Marcoux, Laurent W.; Radjavi, Heydar; Zhang, Yuanhang
    We describe the norm-closures of the set CE of commutators of idempotent operators and the set C-C of differences of idempotent operators on a finite-dimensional complex Hilbert space, as well as characterise the intersection of the closures of these sets with the set K(H) of compact operators acting on an infinite-dimensional complex separable Hilbert space H. Finally, we characterise the closures of the set CP of commutators of orthogonal projections and the set B-B of differences of orthogonal projections acting on a complex separable Hilbert space.
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    Intuitive probabilistic inference in capuchin monkeys
    (Springer Nature, 2017) Tecwyn, Emma C.; Denison, Stephanie; Messer, Emily J. E.; Buchsbaum, Daphna
    The ability to reason about probabilities has ecological relevance for many species. Recent research has shown that both preverbal infants and non-human great apes can make predictions about single-item samples randomly drawn from populations by reasoning about proportions. To further explore the evolutionary origins of this ability, we conducted the first investigation of probabilistic inference in a monkey species (capuchins; Sapajus spp.). Across four experiments, capuchins ( N = 19) were presented with two populations of food items that differed in their relative distribution of preferred and non-preferred items, such that one population was more likely to yield a preferred item. In each trial, capuchins had to select between hidden single-item samples randomly drawn from each population. In Experiment 1 each population was homogeneous so reasoning about proportions was not required; Experiments 2–3 replicated previous probabilistic reasoning research with infants and apes; and Experiment 4 was a novel condition untested in other species, providing an important extension to previous work. Results revealed that at least some capuchins were able to make probabilistic inferences via reasoning about proportions as opposed to simpler quantity heuristics. Performance was relatively poor in Experiment 4, so the possibility remains that capuchins may use quantity-based heuristics in some situations, though further work is required to confirm this. Interestingly, performance was not at ceiling in Experiment 1, which did not involve reasoning about proportions, but did involve sampling. This suggests that the sampling task posed demands in addition to reasoning about proportions, possibly related to inhibitory control, working memory, and/or knowledge of object permanence.