In this thesis, we present several contributions to time **discretization** for highly oscillatory mechanical systems, and to spacetime **discretization** for classical **field** **theories**, both from a common **Lagrangian** variational perspective. The resulting numerical methods have several geometrically desirable properties, including mul- tisymplecticity, conservation of momentum maps via a discrete version of Noether’s theorem, preservation of di ff erential structure and gauge symmetries, lack of spu- rious modes, and excellent long-time energy conservation behavior. Many tradi- tional numerical integrators (such as Runge–Kutta and node-based finite element methods) may fail to preserve one or more of these properties, particularly when simulating dynamical systems with important di ff erential-**geometric** symmetries and structures—as is the case, in particular, with discrete **field** **theories** such as computational electromagnetics and numerical relativity. Like finite element meth- ods, however, the **geometric** methods presented here can be readily applied to unstructured meshes (such as simplicial complexes), with little restriction of mesh topology or geometry.

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= - \ { d ^ K + w U h i) g . , h iŸ - y '‘ (3.49) [53, 54, 55, 56]. Finding the explicit form of the Dirac equation in a given coordinate system is very complicated in general as it requires a knowledge of the Christoffel symbols and the calculation of some lenghty contractions. A judicious choice of tetrad can sim plify things considerably. For this reason one often works in the diagonal tetrad gauge i.e. one chooses three of the tetrad vectors to be parallel to coordinate vectors so that the greatest possible number of tetrad components vanish. In the particular case of light cone coordinates in 3+1 spacetime, even working in the diagonal tetrad gauge, the algebra involved in calculating the **Lagrangian** by the bare hands method is still very involved. We shall therefore obtain the light cone Dirac equation in a different way, by transforming the Dirac **Lagrangian** from Cartesian to light cone coordinates and deriving the **field** equations by the usual variational method. We shall need the **Lagrangian** anyway to discuss the fermion **field** theory. The **Lagrangian** in Minkowski coordinates is

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where, following Ref. [12], we decide to adopt the results concerning the bound-states potential. Even without knowing the detailed calculations of Sec. V, we may point out that, in classical gravity, the Levi-Civita cancellation theorem [58] holds, according to which the N -body **Lagrangian** in general relativity can be always reduced to a **Lagrangian** of N point particles. In other words, it is not necessary to assume that we deal with point particles for simplicity, but the effects of their size get eventually and exactly cancelled. Now the quantum corrections considered in Refs. [13–15] deal precisely with the long- distance Newtonian potential among point particles, and consider three distinct physical settings: scattering, or bound states, or one-particle reducible [12]. We think that, in celestial **mechanics**, the bound states picture is more appropriate for studying stable and unstable equilibrium points. Therefore we set (cf. Sec. V)

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gravitation in which **geometric** objects like nowhere-vanishing vector fields and symplectic forms count as absolute objects. He made his point with the following example. Suppose we have a cosmological model in which there is omnipresent dust, all particles of which are at rest in some Lorentz frame. Pressure-free dust has the stress-energy tensor T ab = ρ U a U b , where the density of the dust particles ρ is defined as the number of particles per unit volume in the unique inertial frame in which the particles are at rest and U a is the four-velocity. In such a universe, the four- velocity would be nowhere-vanishing and would count as an absolute object on Friedman’s definition. That is, there would be a background reference frame in the imaginary model, the rest frame of the dust. Torretti (1984, p. 285) offered another counterexample to the Anderson- Friedman distinction. He formulated a theory of modified Newtonian **mechanics** in which each model has a space of constant non-positive curvature, but different models have different values of curvature. He pointed out that such curvature is undeniably a kind of background-structure, yet escapes the Anderson-Friedman definition of absoluteness. Pitts (2006) presents and challenges these and other counterexamples and he offers a defence of the Anderson-Friedman programme. But he concedes that Einsteinian gravitation may have an absolute object, namely the scalar density obtained by reducing the metric into a conformal metric density and a scalar density. 5

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Layout. This paper consists of five main sections. In Section 2, we describe the **geometric** formulation of ideal fluid flow, and we explain the role played by the diffeomorphism group in this formulation; we summarize some key Lie group theoretic aspects of the diffeomorphism group, and proceed to construct a finite-dimensional approximation of the diffeomorphism group in the manner laid forth by Pavlov and co- authors [14]. In Section 3, we present the theory of Euler-Poincar´ e systems with advected parameters, which provides the variational framework for all of the subsequent continuum **theories** presented in this paper. We then derive a variational temporal **discretization** of the Euler-Poincar´ e equations with advected parameters. In Section 4, we state precisely the **geometric** formulation of ideal fluid flow and proceed to discretize it using the tools developed in Sections 2-3. We then discretize three continuum **theories**: magnetohydrodynamics, nematic liquid crystal flow, and microstretch fluid flow. We present these discretizations as methodically and comprehensively as possible in order to highlight the systematic nature of our approach. In Section 5, we specialize to the case of a cartesian mesh and record the cartesian realizations of the numerical integrators derived in Section 4. Those readers most interested in computational implementation may wish to proceed directly to the end of this section for a concise catalogue of our novel numerical schemes. Finally, in Section 6, we implement our structured integrators on a variety of test cases adapted from the literature. We focus primarily on our MHD integrator since, relative to complex fluid dynamics, the **field** of computational MHD is replete with well-established numerical test cases and existing integrators for comparison. We show nu- merically that our integrators exhibit good long-term energy behavior, preserve certain conserved quantities exactly, respect topological properties of the magnetic **field** that are intrinsic to ideal magnetohydrodynamic flows, and are robust with respect to the spatial and temporal resolution of the grid.

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that the tunnelling out of the vacuum state is described by the instanton solutions of the theory, the imaginary part of the vertex functions i s calculated for the massless theory in th[r]

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The outline of the paper is as follows. In Sec. II, we recall Polchinski ’ s and Wetterich ’ s renormalization group, their duality, and the basic equations of this paper. The beta functions for all couplings are solved locally and globally in Sec. III. Section IV investigates infrared fixed points and convergence-limiting singularities in the complex **field** plane. Asymptotically safe ultraviolet fixed points and strong coupling effects are studied in Sec. V . The sponta- neous breaking of scale symmetry and the Polchinski- Wetterich duality are analyzed in Sec. VI. Section VII investigates the universal eigenperturbations, and the phase diagram including the conformal window with asymptotic safety and regions with first order phase transitions. Section VIII contains a summary of results and implica- tions for asymptotic safety of particle **theories** in four dimensions.

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and i = 0, 1, . . . , D. In [7] the logarithmic scaling behaviour of the von Neumann entropy for these states had already been found. Here we have reproduced and extended that result to the R´ enyi entropy, giving the scaling behaviours (2.24) and (2.25). Popkov et al. were able to obtain their results by exploiting the particularly nice combinatoric features of their chosen state such as the fact that non vanishing eigenvalues of the density matrix are generalised binomial coefficients. For large blocks, a saddle point analysis of these coefficients then leads to the behaviour (2.25). In our case this behaviour follows from Theorem 2, which gives our particular formula for the entanglement entropy based on twist operators [10]. Further, the elementary vectors are very particular linear combinations of our zero-entropy vectors, and thanks to the generality of this formula we have been able to investigate the entanglement entropy of much more general linear combinations (both finite and infinite) and we have shown that the scaling behaviour of the entropy will generally be different depending on the chosen state. Besides the case above, we have explicitly computed this behaviour for the state described in Subsection 5.3, giving (5.31). In this case, the coefficient of the logarithmic term is 2s and is maximal, meaning that no linear combination of permutation symmetric states can have a faster growing entropy than this particular one. We have also explained how other states may be systematically constructed giving coefficients d 2 = 0, 1 2 , 1, 3 2 , . . . , 2s for the logarithmic divergency, and how in these cases the O(1) correction term to this divergency has an interpretation in the context of **geometric** quantum **mechanics**. Finally, we have shown how one may construct states for which the quantity d 2 takes irrational values, with d representing the fractal dimension of some fractal subset of C P 2s . We have provided an explicit construction for s = 1 2 and the Cantor set and a general argument for higher spins and general fractal sets.

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Parcels (“Probably A Really Computationally Efficient **Lagrangian** Simulator”) is a framework for computing La- grangian particle trajectories (http://www.oceanparcels.org, last access: 10 March 2019, Lange and van Sebille, 2017). The main goal of Parcels is to process the continuously increasing number of data generated by the contemporary and future generations of ocean general circulation mod- els (OGCMs). This requires two important features of the model: (1) not to be dependent on one single format of fields and (2) to be able to scale up efficiently to cope with up to petabytes of external data produced by OGCMs. In Lange and van Sebille (2017), the concept of the model was de- scribed and the fundamentals of Parcels v0.9 were stated. Since this version, the model essence has remained the same, but many features were added or improved, leading to the current version 2.0. Among all the developments, our re- search has mainly focused on developing and implementing interpolation schemes to provide the possibility to use a set of fields discretized on various types of grids, from rectilinear to curvilinear in the horizontal direction, with z or s levels in the vertical direction, and using grid staggering with A, B and C Arakawa staggered grids (Arakawa and Lamb, 1977). In particular an interpolation scheme for curvilinear C grids, which was not defined in other **Lagrangian** analysis models, was developed for both z and s levels.

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Abstract. Simulating thimble regularization of lattice **field** theory can be tricky when more than one thimble is to be taken into account. A couple of years ago we proposed a solution for this problem. More recently this solution proved to be effective in the case of 0+1 dimensional QCD. A few lessons we can learnt, including the role of symmetries and general hints on algorithmic solutions.

The second part provides a detailed introduction to the Polyakov-loop Nambu–Jona-Lasinio model [9] for ther- modynamics and mesonic correlations [10] in the phase diagram of quark matter[r]

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One example of such a construction is based on the description of Seiberg-Witten curves of four-dimensional N = 2 supersymmetric gauge **theories** as branes in type IIA string theory and M-theory. This enables us to study the gauge **theories** in strongly-coupled regimes. Spectral networks are another tool for utilizing branes to study non-perturbative regimes of two- and four-dimensional supersymmetric **theories**. Using spectral networks of a Seiberg-Witten theory we can find its BPS spectrum, which is protected from quantum corrections by supersymmetry, and also the BPS spectrum of a related two-dimensional N = (2, 2) theory whose (twisted) superpotential is determined by the Seiberg-Witten curve. When we don’t know the perturbative description of such a theory, its spectrum obtained via spectral networks is a useful piece of information. In this thesis we illustrate these ideas with examples of the use of Seiberg-Witten curves and spectral networks to understand various two- and four-dimensional supersymmetric **theories**.

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A class of nonlocal **field** **theories**, as we dubbed NAQFT, possesses a continuum of massive excitations. As one of the most important features of this class of **theories**, we studied the role of the continuum modes in detail. The path integral formulation of this theory was derived and led to a dual picture in terms of local fields. The dual picture highlights how the continuum modes of the nonlocal **field** behave. We have also derived the Feynman rules for evaluating the S-matrix amplitudes using the dual picture.

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We present a systematic derivation for a general formula for the n-point coupling constant valid for affine Toda **theories** related to any simple Lie algebra g. All n-point couplings with n ≥ 4 are completely determined in terms of the masses and the three-point couplings. A general fusing rule, formulated in the root space of the Lie algebra, is derived for all n-point couplings.

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We base our discussion of string theory on the lecture notes of R. J. Szabo [21]. String theory is still work in progress, and a thorough discussion of the relation of string theory to the approach in this paper is elusive so far. However, there are ﬁve diﬀerent consistent formulations of string theory that are commonly seen as perturbative expansions of a unique underlying theory (M-theory), which is however not well understood yet. The ﬁve **theories** are related by dualities that map perturbative states in one theory to non-perturbative states in another theory.

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We investigate the thermodynamic Bethe ansatz (TBA) equations for a system of particles which dynamically interacts via the scattering matrix of affine Toda **field** theory and whose statistical interaction is of a general Haldane type. Up to the first leading order, we provide general approximated analytical expressions for the solutions of these equations from which we derive general formulae for the ultraviolet scaling functions for **theories** in which the underlying Lie algebra is simply laced. For several explicit models we compare the quality of the approximated analytical solutions against the numerical solutions. We address the question of existence and uniqueness of the solutions of the TBA-equations, derive precise error estimates and determine the rate of convergence for the applied numerical procedure. A general expression for the Fourier transformed kernels of the TBA-equations allows to derive the related Y-systems and a reformulation of the equations into a universal form.

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[8] J. Donea, A. Huerta, J.P. Ponthot, A. Rodr´ıguez-Ferran, Arbitrary **Lagrangian**– Eulerian Methods, Encyclopedia of Computational **Mechanics**, Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes. Volume 1: Fundamentals., 1 (2004). [9] E. Kuhl, H. Askes, P. Steinmann, An ALE formulation based on spatial and material settings of continuum **mechanics**. Part 1: Generic hyperelastic formulation, Computer Methods in Applied **Mechanics** and Engineering, 193 (2004) 4207-4222. [10] H. Askes, E. Kuhl, P. Steinmann, An ALE formulation based on spatial and material settings of continuum **mechanics**. Part 2: Classification and applications, Computer Methods in Applied **Mechanics** and Engineering, 193 (2004) 4223-4245. [11] A. Legay, J. Chessa, T. Belytschko, An Eulerian–**Lagrangian** method for fluid– structure interaction based on level sets, Computer Methods in Applied **Mechanics** and Engineering, 195 2070-2087.

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However, twentieth century physics developed in a direction that formally obscures this under- lying symmetry of Maxwell’s equations. Recasting electromagnetism as a gauge theory (that is, identifying the electromagnetic **field** as a connection A on a principal U (1) bundle over spacetime), duality ceases to be manifest since the 2-form F = dA and its Hodge dual ∗dA play very dissimilar roles in the formalism. In particular, the inclusion of electric sources is handled very differently from that of magnetic sources: electric sources become Wilson line order operators, included as a source term in the action, while magnetic sources become t’Hooft line disorder operators, included by imposing a prescribed singularity in the space of **field** configurations. Electric-magnetic duality turns out to extend to the quantum theory of a U (1) connection and the equivalence, though still provable [8], becomes even more hidden by the formalism.

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Kac-Moody Lie algebras were discovered independently by Victor G. Kac and Robert V. Moody around 1968. These algebras are infinite dimensional analogs of finite dimensional semisimple Lie algebras. There are three types of Kac-Moody Lie algebras: finite type, affine type, and indefinite type. These algebras, especially affine Lie algebras, have various applica- tions in physics and mathematics. In 1985, Michio Jimbo and Vladimir G. Drinfeld introduced the notion of the quantum group which are deformations of the universal enveloping algebras of symmetrizable Kac-Moody Lie algebras. In 1988, George Lusztig showed that the represen- tation theory of a Kac-Moody Lie algebra is parallel to that of its quantum group in the generic case. In 1990, Masaki Kashiwara developed the crystal basis as a nice combinatorial tool to study the irreducible highest weight modules over a quantum group. Then, in 1999, the notion of **geometric** crystal is introduced by Arkady Berenstein and David Kazhdan as a **geometric** analog to Kashiwara’s crystal (or algebraic crystal). A remarkable relation between positive **geometric** crystals and algebraic crystals is the ultra-**discretization** functor U D between them. Applying this functor, positive rational functions are transferred to piecewise linear functions. In 2008, Masaki Kashiwara, Toshiki Nakashima and Masato Okado gave a conjecture that for each affine Lie algebra g and each Dynkin index i ∈ I \ { 0 } , there exists a posi- tive **geometric** crystal V ( g ) = ( X, { e i } i ∈ I , { γ i } i ∈ I , { ε i } i ∈ I ) whose ultra-**discretization** U D(V )

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nonperturbative parameter that we determined from 𝜒 cJ (1P) decay rates. ▸ This is the first prediction of heavy[r]

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