Academic literature on the topic 'Hurwitz metrics'

In this paper, we introduce a new class of integrable systems, naturally associated to Hurwitz spaces (spaces of meromorphic functions over Riemann surfaces). The critical values of the meromorphic functions play the role of ‘times’. Our systems give a natural generalization of the Ernst equation; in genus zero, they realize the scheme of deformation of integrable systems proposed by Burtsev, Mikhailov and Zakharov. We show that any solution of these systems in rank 1 defines a flat diagonal metric (Darboux–Egoroff metric) together with a class of corresponding systems of hydrodynamic type and their solutions.

By using tensor analysis, we find a connection between normed algebras and the parallelizability of the spheres S 1, S 3 and S 7. In this process, we discovered the analog of Hurwitz theorem for curved spaces and a geometrical unified formalism for the metric and the torsion. In order to achieve these goals we first develop a proof of Hurwitz theorem based on tensor analysis. It turns out that in contrast to the doubling procedure and Clifford algebra mechanism, our proof is entirely based on tensor algebra applied to the normed algebra condition. From the tersor analysis point of view our proof is straightforward and short. We also discuss a possible connection between our formalism and the Cayley–Dickson algebras and Hopf maps.

Decimetric to metric domal stromatolites with constituent ministromatolites characterize reddish, 13C-enriched dolostones in the Watterson Formation of the Quartzite Lake area west of Hudson Bay. They provide paleontologic support for a correlation with the only other known early Paleoproterozoic stromatolite occurrences in North America: the Kona Formation of Michigan, and the Nash Formation in southern Wyoming. They also are similar to stromatolites in probable coeval Jatulian carbonates in Karelia on the Baltic Shield, and possibly to stromatolites in the Hutuo Group in China.

Abstract The diversity of T cell receptors (TCRs) becomes increasingly restricted with advancing stage in renal cell carcinoma (RCC), with individual T cell clones becoming dramatically expanded. Further, prior studies have suggested that TCR clonality (i.e. less diversity) could be associated with improved clinical response to anti-PD-1 treatment. To further interrogate the TCR landscape of advanced RCC and its relation to both T cell phenotype and clinical outcomes, single-cell TCR sequencing (scTCR-seq) was performed in parallel with single-cell RNA sequencing (scRNA-seq) for eligible patients with clear cell or non-clear cell metastatic RCC both pre- (baseline) or post- (progression) treatment with front-line nivolumab monotherapy as part of the HCRN GU16-260 trial (NCT03117309). Clonotype assignments and diversity metrics for 19 tumor samples were determined using VDJdive (R/Bioconductor). Overall, the top 10 clonotypes accounted for 12.3% of all T cells (range by sample: 4.2-34.6%). Of note, 5 of 16 (31%) samples with at least 100 T cells displayed substantial clonotype expansion, with the top 10 clonotypes accounting for at least 20% of tumor-infiltrating T cells. Average TCR diversity (normalized Shannon entropy) was 0.55 (range: 0.38-0.91) for all 19 samples. There were no significant differences in T cell diversity both between baseline (n=10) and post-treatment (n=9) samples (p= 0.66) or between clear cell (n=14) and non-clear cell (n=5) histologies (p= 0.75). To assess the relationship between T cell specificity and phenotypic state, we investigated the distribution of the most expanded clonotypes along a trajectory of T cells. Previous trajectory analysis of T cell populations revealed a bifurcating structure with naïve T cells at the root and lineages terminating in either terminally exhausted CD8+ T cells or SLAMF7+ CD8+ T cells (Braun, ASCO GU, 2023). Of the top 20 most expanded clonotypes across all T cells, 95% were distributed almost exclusively in the terminally exhausted lineage. The TCR diversity decreased along each lineage, from naïve T cell cluster at the root (TCR diversity = 0.61) to either the terminally exhausted CD8+ T cell cluster (0.42) or the SLAMF7+ CD8+ T cell cluster (0.56). There were no significant differences in TCR diversity between patients with progressive disease (n=6) versus complete/partial response (n= 6), (p= 0.96). Similarly, there was no association between TCR diversity and either progression-free (p= 0.41) or overall (p=0.39) survival. Altogether, these data reaffirm the presence of clonal expansion in advanced RCC, with the degree varying by sample. The localization of most expanded clonotypes to a single T cell lineage suggests a low degree of phenotypic plasticity. Enrichment of these clonotypes along an exhausted lineage is consistent with prior findings of lower TCR diversity in terminally exhausted CD8+ T cells. Future directions will focus on inferring the tumor specificity of infiltrating T cells along the diverging lineages. Citation Format: Miya B. Hugaboom, Neil Ruthen, Opeyemi Jegede, Nicholas R. Schindler, Lena Wirth, Sasha Kyrysyuk, David F. McDermott, Elizabeth R. Plimack, Jeffrey A. Sosman, Naomi B. Haas, Michael E. Hurwitz, Hans J. Hammers, Sabina Signoretti, Michael B. Atkins, Catherine J. Wu, David A. Braun, Kelly Street. T cell clonotype expansion is common in advanced renal cell carcinoma but is not associated with altered response to PD-1 blockade [abstract]. In: Proceedings of the AACR Special Conference: Advances in Kidney Cancer Research; 2023 Jun 24-27; Austin, Texas. Philadelphia (PA): AACR; Cancer Res 2023;83(16 Suppl):Abstract nr B013.

We present a new distributionally robust optimization model called robust stochastic optimization (RSO), which unifies both scenario-tree-based stochastic linear optimization and distributionally robust optimization in a practicable framework that can be solved using the state-of-the-art commercial optimization solvers. We also develop a new algebraic modeling package, Robust Stochastic Optimization Made Easy (RSOME), to facilitate the implementation of RSO models. The model of uncertainty incorporates both discrete and continuous random variables, typically assumed in scenario-tree-based stochastic linear optimization and distributionally robust optimization, respectively. To address the nonanticipativity of recourse decisions, we introduce the event-wise recourse adaptations, which integrate the scenario-tree adaptation originating from stochastic linear optimization and the affine adaptation popularized in distributionally robust optimization. Our proposed event-wise ambiguity set is rich enough to capture traditional statistic-based ambiguity sets with convex generalized moments, mixture distribution, φ-divergence, Wasserstein (Kantorovich-Rubinstein) metric, and also inspire machine-learning-based ones using techniques such as K-means clustering and classification and regression trees. Several interesting RSO models, including optimizing over the Hurwicz criterion and two-stage problems over Wasserstein ambiguity sets, are provided. This paper was accepted by David Simchi-Levi, optimization.

The main aim of this thesis is to explain the behaviour of some conformal metrics and invariants near a smooth boundary point of a domain in the complex plane. We will be interested in the invariants associated to the Carathéodory metric such as its higher-order curvatures that were introduced by Burbea and the Aumann-Carathéodory rigidity constant, the Sugawa metric and the Hurwitz metric. The basic technical step in all these is the method of scaling the domain near a smooth boundary point. To estimate the higher-order curvatures using scaling, we generalize an old theorem of Suita on the real analyticity of the Carathéodory metric on planar domains and in the process, we show convergence of the Szeg˝o and Garabedian kernels as well. By using similar ideas we also show that the Aumann-Carathéodory rigidity constant converges to 1 near smooth boundary points. Next on the line is a conformal metric defined using holomorphic quadratic differentials. Thiswas done by T. Sugawa andwe will refer to this as the Sugawa metric. It is shown that this metric is uniformly comparable to the quasi-hyperbolic metric on a smoothly bounded domain. We also study the Hurwitz metric that was introduced by D. Minda. Its construction is similar to the Kobayashi metric but the essential difference lies in the class of holomorphic maps that are considered in its definition. We show that this metric is continuous and also strengthen Minda’s theorem about its comparability with the quasi-hyperbolic metric by estimating the constants in a more natural manner. Finally, we get some weak estimates on the generalized upper and lower curvatures of the Sugawa and Hurwitz metrics.

In the first chapter of this report, our aim is to introduce harmonic maps between Riemann surfaces using the Energy integral of a map. Once we have the desired prerequisites, we move on to show how to continuously deform a given map to a harmonic map (i.e., find a harmonic map in its homotopy class). We follow J¨urgen Jost’s approach using classical potential theory techniques. Subsequently, we analyze the additional conditions needed to ensure a certain uniqueness property of harmonic maps within a given homotopy class. In conclusion, we look at a couple of applications of what we have shown thus far and we find a neat proof of a slightly weaker version of Hurwitz’s Automorphism Theorem. In the second chapter, we introduce the concept of minimal surfaces. After exploring a few examples, we mathematically formulate Plateau’s problem regarding the existence of a soap film spanning each closed, simple wire frame and discuss a solution. In conclusion, a partial result (due to Rad´o) regarding the uniqueness of such a soap film is discussed.

In the first chapter of this report, our aim is to introduce harmonic maps between Riemann surfaces using the Energy integral of a map. Once we have the desired prerequisites, we move on to show how to continuously deform a given map to a harmonic map (i.e., find a harmonic map in its homotopy class). We follow J¨urgen Jost’s approach using classical potential theory techniques. Subsequently, we analyze the additional conditions needed to ensure a certain uniqueness property of harmonic maps within a given homotopy class. In conclusion, we look at a couple of applications of what we have shown thus far and we find a neat proof of a slightly weaker version of Hurwitz’s Automorphism Theorem. In the second chapter, we introduce the concept of minimal surfaces. After exploring a few examples, we mathematically formulate Plateau’s problem regarding the existence of a soap film spanning each closed, simple wire frame and discuss a solution. In conclusion, a partial result (due to Rad´o) regarding the uniqueness of such a soap film is discussed.