2,513 research outputs found

    Symmetries of Riemann surfaces and magnetic monopoles

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    This thesis studies, broadly, the role of symmetry in elucidating structure. In particular, I investigate the role that automorphisms of algebraic curves play in three specific contexts; determining the orbits of theta characteristics, influencing the geometry of the highly-symmetric Bring’s curve, and in constructing magnetic monopole solutions. On theta characteristics, I show how to turn questions on the existence of invariant characteristics into questions of group cohomology, compute comprehensive tables of orbit decompositions for curves of genus 9 or less, and prove results on the existence of infinite families of curves with invariant characteristics. On Bring’s curve, I identify key points with geometric significance on the curve, completely determine the structure of the quotients by subgroups of automorphisms, finding new elliptic curves in the process, and identify the unique invariant theta characteristic on the curve. With respect to monopoles, I elucidate the role that the Hitchin conditions play in determining monopole spectral curves, the relation between these conditions and the automorphism group of the curve, and I develop the theory of computing Nahm data of symmetric monopoles. As such I classify all 3-monopoles whose Nahm data may be solved for in terms of elliptic functions

    The infrared structure of perturbative gauge theories

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    Infrared divergences in the perturbative expansion of gauge theory amplitudes and cross sections have been a focus of theoretical investigations for almost a century. New insights still continue to emerge, as higher perturbative orders are explored, and high-precision phenomenological applications demand an ever more refined understanding. This review aims to provide a pedagogical overview of the subject. We briefly cover some of the early historical results, we provide some simple examples of low-order applications in the context of perturbative QCD, and discuss the necessary tools to extend these results to all perturbative orders. Finally, we describe recent developments concerning the calculation of soft anomalous dimensions in multi-particle scattering amplitudes at high orders, and we provide a brief introduction to the very active field of infrared subtraction for the calculation of differential distributions at colliders. © 2022 Elsevier B.V

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    A First Course in Causal Inference

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    I developed the lecture notes based on my ``Causal Inference'' course at the University of California Berkeley over the past seven years. Since half of the students were undergraduates, my lecture notes only require basic knowledge of probability theory, statistical inference, and linear and logistic regressions

    Mirror symmetry for Dubrovin-Zhang Frobenius manifolds

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    Frobenius manifolds were formally defined by Boris Dubrovin in the early 1990s, and serve as a bridge between a priori very different fields of mathematics such as integrable systems theory, enumerative geometry, singularity theory, and mathematical physics. This thesis concerns, in particular, a specific class of Frobenius manifolds constructed on the orbit space of an extension of the affine Weyl group defined by Dubrovin together with Youjin Zhang. Here, we find Landau-Ginzburg superpotentials, or B-model mirrors, for these Frobenius structures by considering the characteristic equation for Lax operators of relativistic Toda chains as proposed by Andrea Brini. As a bonus, the results open up various applications in topology, integrable hierarchies, and Gromov-Witten theory, making interesting research questions in these areas more accessible. Some such applications are considered in this thesis. The form of the determinant of the Saito metric on discriminant strata is investigated, applications to the combinatorics of Lyashko-Looijenga maps are given, and investigations into the integrable systems theoretic and enumerative geometric applications are commenced

    Flat coordinates of algebraic Frobenius manifolds in small dimensions

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    Orbit spaces of the reflection representation of finite irreducible Coxeter groups provide polynomial Frobenius manifolds. Flat coordinates of the Frobenius metric η\eta are Saito polynomials which are distinguished basic invariants of the Coxeter group. Algebraic Frobenius manifolds are typically related to quasi-Coxeter conjugacy classes in finite Coxeter groups. We find explicit relations between flat coordinates of the Frobenius metric η\eta and flat coordinates of the intersection form gg for most known examples of algebraic Frobenius manifolds up to dimension 4. In all the cases, flat coordinates of the metric η\eta appear to be algebraic functions on the orbit space of the Coxeter group.Comment: 53 page

    An Intelligent Time and Performance Efficient Algorithm for Aircraft Design Optimization

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    Die Optimierung des Flugzeugentwurfs erfordert die Beherrschung der komplexen Zusammenhänge mehrerer Disziplinen. Trotz seiner Abhängigkeit von einer Vielzahl unabhängiger Variablen zeichnet sich dieses komplexe Entwurfsproblem durch starke indirekte Verbindungen und eine daraus resultierende geringe Anzahl lokaler Minima aus. Kürzlich entwickelte intelligente Methoden, die auf selbstlernenden Algorithmen basieren, ermutigten die Suche nach einer diesem Bereich zugeordneten neuen Methode. Tatsächlich wird der in dieser Arbeit entwickelte Hybrid-Algorithmus (Cavus) auf zwei Hauptdesignfälle im Luft- und Raumfahrtbereich angewendet: Flugzeugentwurf- und Flugbahnoptimierung. Der implementierte neue Ansatz ist in der Lage, die Anzahl der Versuchspunkte ohne große Kompromisse zu reduzieren. Die Trendanalyse zeigt, dass der Cavus-Algorithmus für die komplexen Designprobleme, mit einer proportionalen Anzahl von Prüfpunkten konservativer ist, um die erfolgreichen Muster zu finden. Aircraft Design Optimization requires mastering of the complex interrelationships of multiple disciplines. Despite its dependency on a diverse number of independent variables, this complex design problem has favourable nature as having strong indirect links and as a result a low number of local minimums. Recently developed intelligent methods that are based on self-learning algorithms encouraged finding a new method dedicated to this area. Indeed, the hybrid (Cavus) algorithm developed in this thesis is applied two main design cases in aerospace area: aircraft design optimization and trajectory optimization. The implemented new approach is capable of reducing the number of trial points without much compromise. The trend analysis shows that, for the complex design problems the Cavus algorithm is more conservative with a proportional number of trial points in finding the successful patterns

    On representation-theoretic properties of fermionic fields in de Sitter spacetime and symmetries underlying the conservation of the electromagnetic zilches

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    This journal-style thesis presents chapters 2-6 in a format suitable for peer-reviewed publication. In chapter 2, we study the quantum spinor field on NN-dimensional de Sitter spacetime (dSNdS_{N} ) with (N1)(N − 1)-sphere (SN1S^{N−1}) spatial sections. We construct the mode solutions and study their transformation properties under the de Sitter (dS) algebra, spin(N,1)(N, 1). We reproduce the expression for the massless spinor Wightman two-point function using the mode-sum method. Then, taking advantage of the maximal symmetry of dSNdS_{N} , we construct the massive Wightman two-point function. In chapter 3, we construct the dictionary between the spaces of mode solutions for totally symmetric spin-s=3/2,5/2s = 3/2, 5/2 tensor-spinors with any mass parameter on dSNdS_{N} (N3N \geq 3) and Unitary Irreducible Representations (UIRs) of spin(N,1)(N, 1). Remarkably, we find that the strictly massless spin-3/2 field, as well as the strictly and partially massless spin-5/2 fields on dSNdS_{N}, are not unitary unless N=4N = 4. Chapter 4 provides a technical explanation for the results of chapter 3 by investigating the (non-)existence of positive-definite, dS invariant scalar products for the mode solutions. In chapter 5, we uncover a ‘conformal-like’ spinspin(4, 2) symmetry for strictly massless spin-s3/2s \geq 3/2 tensor-spinors on dS4dS_{4}. We also show that the mode solutions form UIRs of not only the dS algebra but also of spin(4,2)(4, 2). In chapter 6, we shift focus to the ‘zilches’, a set of little-known conserved quantities for the free electromagnetic (EM) field in four-dimensional Minkowski spacetime. We present, for the first time, the derivation of all zilch conservation laws from ‘zilch symmetries’ of the standard EM action using Noether’s theorem. We also show that the zilch symmetries belong to the enveloping algebra of a "hidden" invariance algebra of free Maxwell’s equations in potential form

    Numerical methods for computing the discrete and continuous Laplace transforms

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    We propose a numerical method to spline-interpolate discrete signals and then apply the integral transforms to the corresponding analytical spline functions. This represents a robust and computationally efficient technique for estimating the Laplace transform for noisy data. We revisited a Meijer-G symbolic approach to compute the Laplace transform and alternative approaches to extend canonical observed time-series. A discrete quantization scheme provides the foundation for rapid and reliable estimation of the inverse Laplace transform. We derive theoretic estimates for the inverse Laplace transform of analytic functions and demonstrate empirical results validating the algorithmic performance using observed and simulated data. We also introduce a generalization of the Laplace transform in higher dimensional space-time. We tested the discrete LT algorithm on data sampled from analytic functions with known exact Laplace transforms. The validation of the discrete ILT involves using complex functions with known analytic ILTs
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