14,376 research outputs found
Wave-function parametrization of a probability measure
We show that the unitary operator on a separable Hilbert space is a
parametrization of any conditional probability measure in a standard measure
space. We propose unitary inference, a generalization of Bayesian inference. We
study implications for classical statistical mechanics.Comment: 15 pages, v3: Basis change of prior wave-function discussed and other
minor improvement
Lie-Poisson gauge theories and -Minkowski electrodynamics
We consider a Poisson gauge theory with a generic Poisson structure of Lie
algebraic type. We prove an important identity, which allows to obtain simple
and manifestly gauge-covariant expressions for the Euler-Lagrange equations of
motion, the Bianchi and the Noether identities. We discuss the non-Lagrangian
equations of motion, and apply our findings to the -Minkowski case. We
construct a family of exact solutions of the deformed Maxwell equations in the
vacuum. In the classical limit, these solutions recover plane waves with
left-handed and right-handed circular polarization, being classical
counterparts of photons. The deformed dispersion relation appears to be
nontrivial.Comment: 20 page
Rank-based linkage I: triplet comparisons and oriented simplicial complexes
Rank-based linkage is a new tool for summarizing a collection of objects
according to their relationships. These objects are not mapped to vectors, and
``similarity'' between objects need be neither numerical nor symmetrical. All
an object needs to do is rank nearby objects by similarity to itself, using a
Comparator which is transitive, but need not be consistent with any metric on
the whole set. Call this a ranking system on . Rank-based linkage is applied
to the -nearest neighbor digraph derived from a ranking system. Computations
occur on a 2-dimensional abstract oriented simplicial complex whose faces are
among the points, edges, and triangles of the line graph of the undirected
-nearest neighbor graph on . In steps it builds an
edge-weighted linkage graph where
is called the in-sway between objects and . Take to be
the links whose in-sway is at least , and partition into components of
the graph , for varying . Rank-based linkage is a
functor from a category of out-ordered digraphs to a category of partitioned
sets, with the practical consequence that augmenting the set of objects in a
rank-respectful way gives a fresh clustering which does not ``rip apart`` the
previous one. The same holds for single linkage clustering in the metric space
context, but not for typical optimization-based methods. Open combinatorial
problems are presented in the last section.Comment: 37 pages, 12 figure
Translation of: Sur des familles remarquables d’hypersurfaces isoparamétriques dans les espaces sphériques, Mathematische Zeitschrift 45, 335–367 (1939), by Élie Cartan.
This is an English translation of the article Sur des familles remarquables d’hypersurfaces isoparamétriques dans les espaces sphériques, which was originally published in Mathematische Zeitschrift 45, 335–367 (1939), by Élie Cartan.
A note from Thomas E. Cecil, translator: This is an unofficial translation of the original paper which was written in French. All references should be made to the original paper.
Mathematics Subject Classification Numbers: 53B25, 53C40, 53C4
3d mirror symmetry of braided tensor categories
We study the braided tensor structure of line operators in the topological A
and B twists of abelian 3d gauge theories, as accessed via
boundary vertex operator algebras (VOA's). We focus exclusively on abelian
theories. We first find a non-perturbative completion of boundary VOA's in the
B twist, which start out as certain affine Lie superalebras; and we construct
free-field realizations of both A and B-twist VOA's, finding an interesting
interplay with the symmetry fractionalization group of bulk theories. We use
the free-field realizations to establish an isomorphism between A and B VOA's
related by 3d mirror symmetry. Turning to line operators, we extend previous
physical classifications of line operators to include new monodromy defects and
bound states. We also outline a mechanism by which continuous global symmetries
in a physical theory are promoted to higher symmetries in a topological twist
-- in our case, these are infinite one-form symmetries, related to boundary
spectral flow, which structure the categories of lines and control abelian
gauging. Finally, we establish the existence of braided tensor structure on
categories of line operators, viewed as non-semisimple categories of modules
for boundary VOA's. In the A twist, we obtain the categories by extending
modules of symplectic boson VOA's, corresponding to gauging free
hypermultiplets; in the B twist, we instead extend Kazhdan-Lusztig categories
for affine Lie superalgebras. We prove braided tensor equivalences among the
categories of 3d-mirror theories. All results on VOA's and their module
categories are mathematically rigorous; they rely strongly on recently
developed techniques to access non-semisimple extensions.Comment: 158 pages, comments welcome
Multiscale structural optimisation with concurrent coupling between scales
A robust three-dimensional multiscale topology optimisation framework with concurrent coupling between scales is presented. Concurrent coupling ensures that only the microscale data required to evaluate the macroscale model during each iteration of optimisation is collected and results in considerable computational savings. This represents the principal novelty of the framework and permits a previously intractable number of design variables to be used in the parametrisation of the microscale geometry, which in turn enables accessibility to a greater range of mechanical point properties during optimisation. Additionally, the microscale data collected during optimisation is stored in a re-usable database, further reducing the computational expense of subsequent iterations or entirely new optimisation problems. Application of this methodology enables structures with precise functionally-graded mechanical properties over two-scales to be derived, which satisfy one or multiple functional objectives. For all applications of the framework presented within this thesis, only a small fraction of the microstructure database is required to derive the optimised multiscale solutions, which demonstrates a significant reduction in the computational expense of optimisation in comparison to contemporary sequential frameworks.
The derivation and integration of novel additive manufacturing constraints for open-walled microstructures within the concurrently coupled multiscale topology optimisation framework is also presented. Problematic fabrication features are discouraged through the application of an augmented projection filter and two relaxed binary integral constraints, which prohibit the formation of unsupported members, isolated assemblies of overhanging members and slender members during optimisation. Through the application of these constraints, it is possible to derive self-supporting, hierarchical structures with varying topology, suitable for fabrication through additive manufacturing processes.Open Acces
Moduli Stabilisation and the Statistics of Low-Energy Physics in the String Landscape
In this thesis we present a detailed analysis of the statistical properties of the type IIB flux landscape of string theory. We focus primarily on models constructed via the Large Volume Scenario (LVS) and KKLT and study the distribution of various phenomenologically relevant quantities. First, we compare our considerations with previous results and point out the importance of Kähler moduli stabilisation, which has been neglected in this context so far. We perform different moduli stabilisation procedures and compare the resulting distributions. To this end, we derive the expressions for the gravitino mass, various quantities related to axion physics and other phenomenologically interesting quantities in terms of the fundamental flux dependent quantities , and , the parameter which specifies the nature of the non-perturbative effects. Exploiting our knowledge of the distribution of these fundamental parameters, we can derive a distribution for all the quantities we are interested in. For models that are stabilised via LVS we find a logarithmic distribution, whereas for KKLT and perturbatively stabilised models we find a power-law distribution. We continue by investigating the statistical significance of a newly found class of KKLT vacua and present a search algorithm for such constructions. We conclude by presenting an application of our findings. Given the mild preference for higher scale supersymmetry breaking, we present a model of the early universe, which allows for additional periods of early matter domination and ultimately leads to rather sharp predictions for the dark matter mass in this model. We find the dark matter mass to be in the very heavy range
Exploring the Structure of Scattering Amplitudes in Quantum Field Theory: Scattering Equations, On-Shell Diagrams and Ambitwistor String Models in Gauge Theory and Gravity
In this thesis I analyse the structure of scattering amplitudes in super-symmetric gauge and gravitational theories in four dimensional spacetime, starting with a detailed review of background material accessible to a non-expert. I then analyse the 4D scattering equations, developing the theory of how they can be used to express scattering amplitudes at tree level. I go on to explain how the equations can be solved numerically using a Monte Carlo algorithm, and introduce my Mathematica package treeamps4dJAF which performs these calculations. Next I analyse the relation between the 4D scattering equations and on-shell diagrams in N = 4 super Yang-Mills, which provides a new perspective on the tree level amplitudes of the theory. I apply a similar analysis to N = 8 supergravity, developing the theory of on-shell diagrams to derive new Grassmannian integral formulae for the amplitudes of the theory. In both theories I derive a new worldsheet expression for the 4 point one loop amplitude supported on 4D scattering equations. Finally I use 4D ambitwistor string theory to analyse scattering amplitudes in N = 4 conformal supergravity, deriving new worldsheet formulae for both plane wave and non-plane wave amplitudes supported on 4D scattering equations. I introduce a new prescription to calculate the derivatives of on-shell variables with respect to momenta, and I use this to show that certain non-plane wave amplitudes can be calculated as momentum derivatives of amplitudes with plane wave states
Trigonometric ∨-systems and solutions of WDVV and related equations
This thesis contains three parts related to trigonometric solutions of Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations.
In the first part of the thesis we consider a class of trigonometric solutions of WDVV equations determined by collections A of covectors with multiplicities. This class of solutions involves an extra variable which makes them non trivial already for planar collections. These solutions have the general form
F = ∑     cαf(α(x)) + Q,                                   (0.1)
      α∈A
where cα ∈ C are multiplicity parameters and Q is a cubic polynomial in x = (x1, . . . , xN) and additional variable y, and f is the function of a single variable z satisfying f′′′(z) = cot z. We show that such solutions can be restricted to special subspaces to produce new solutions of the same type. We find new solutions given by restrictions of root systems, as well as examples which are not of this form. Further, we consider a closely related notion of a trigonometric ∨-system, and we show that its subsystems are also trigonometric ∨-systems. While reviewing the root system case we determine a version of generalised Coxeter number for the exterior square of the reflection representation of a Weyl group. We give a list of all the known trigonometric ∨-systems on the plane.
In the second part of the thesis, we consider solutions of WDVV equations in N-dimensional space (without extra variable), which are of the form (0.1) with Q = 0. Such class of solutions does not exist in general even for the case of root system A and invariant multiplicities cα. However, it is known to exist for the root system BN and specific choice of invariant multiplicities [33]. We generalize this solution to a multiparameter family so that the underlying configuration A is the root system BCN. These BCN type solutions of WDVV equations are found by applying restrictions to the known solutions of the commutativity equations and by relating commutativity equations with WDVV equations for the corresponding prepotential. We apply these solutions to define N = 4 supersymmetric mechanical systems.
In the third part of the thesis we reveal the relation between the set of WDVV equations and the set of the commutativity equations for an arbitrary function F. We reformulate it as the existence of the identity vector field for the natural associative multiplication associated with a solution F of commutativity equations. We give explicit formulas of the identity vector field corresponding to root systems for all the cases when commutativity equations are known to be satisfied. We also get new solutions of WDVV equations related to root system F4
Gauge invariant coefficients in perturbative quantum gravity
Perturbative quantum gravity can be studied in many ways. A traditional approach is to apply covariant quantization schemes to the Einstein-Hilbert action and use heat kernel methods, as pioneered by DeWitt. An alternative approach is to consider the graviton as arising from the first quantization of particle actions, following the same methods used in string theory. An interesting model to describe the graviton is based on the so-called N = 4 spinning particle, which has been used recently to study perturbative properties of quantum gravity, allowing in particular for the calculation of certain gauge-invariant coefficients. The latter are related to the counterterms that renormalize the one-loop effective action of pure quantum gravity with a cosmological constant. Such coefficients have already been tested in D = 4 dimensions. Here we study the general case of arbitrary D. We derive the gauge-invariant coefficients —the simplest one being the number of physical degrees of freedom of the graviton—using the traditional heat kernel method. We compare them with the ones obtained by using the N = 4 spinning particle and discover that the latter fails to reproduce some of those coefficients at arbitrary dimension, suggesting the need of improving that first quantized model. This constitutes a first original result of this thesis. In the second part, we try to find an alternative worldline path integral treatment of the heat kernel, extending a previous worldline construction that was tailored to 4 dimensions only. We succeed in finding suitable worldline actions for the gauge-fixed graviton fluctuations and related ghosts. The action for the graviton fluctuations that we construct reproduces the expected Hamiltonian but does not seem to admit a perturbative path integral treatment
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