856 research outputs found
Singularities of Functions on the Martinet Plane, Constrained Hamiltonian Systems and Singular Lagrangians
We consider here the analytic classification of pairs where
is a germ of a 2-form on the plane and is a quasihomogeneous
function germ with isolated singularities. We consider only the case where
is singular, i.e. it vanishes non-degenerately along a smooth line
(Martinet case) and the function is such that the pair
defines an isolated boundary singularity. In analogy with the
ordinary case (for symplectic forms on the plane) we show that the moduli in
the classification problem are analytic functions of 1-variable and that their
number is exactly equal to the Milnor number of the corresponding boundary
singularity. Moreover we derive a normal form for the pair
involving exactly these functional invariants. Finally we give an application
of the results in the theory of constrained Hamiltonian systems, related to the
motion of charged particles in the quantisation limit in an electromagnetic
field, which in turn leads to a list of normal forms of generic singular
Lagrangians (of first order in the velocities) on the plane
High frequency trading and asymptotics for small risk aversion in a Markov renewal model
We study a an optimal high frequency trading problem within a market
microstructure model designed to be a good compromise between accuracy and
tractability. The stock price is driven by a Markov Renewal Process (MRP),
while market orders arrive in the limit order book via a point process
correlated with the stock price itself. In this framework, we can reproduce the
adverse selection risk, appearing in two different forms: the usual one due to
big market orders impacting the stock price and penalizing the agent, and the
weak one due to small market orders and reducing the probability of a
profitable execution. We solve the market making problem by stochastic control
techniques in this semi-Markov model. In the no risk-aversion case, we provide
explicit formula for the optimal controls and characterize the value function
as a simple linear PDE. In the general case, we derive the optimal controls and
the value function in terms of the previous result, and illustrate how the risk
aversion influences the trader strategy and her expected gain. Finally, by
using a perturbation method, approximate optimal controls for small risk
aversions are explicitly computed in terms of two simple PDE's, reducing
drastically the computational cost and enlightening the financial
interpretation of the results.Comment: 30 pages, new asymptotic results, typos corrected, new
bibliographical reference
Landscape of superconducting membranes
The AdS/CFT correspondence may connect the landscape of string vacua and the
`atomic landscape' of condensed matter physics. We study the stability of a
landscape of IR fixed points of N=2 large N gauge theories in 2+1 dimensions,
dual to Sasaki-Einstein compactifications of M theory, towards a
superconducting state. By exhibiting instabilities of charged black holes in
these compactifications, we show that many of these theories have charged
operators that condense when the theory is placed at a finite chemical
potential. We compute a statistical distribution of critical superconducting
temperatures for a subset of these theories. With a chemical potential of one
milliVolt, we find critical temperatures ranging between 0.24 and 165 degrees
Kelvin.Comment: 1+34 pages. 3 figures. v2 references added, typos fixe
A general phase-field model for fatigue failure in brittle and ductile solids
In this work, the phase-field approach to fracture is extended to model fatigue failure in high- and low-cycle regime. The fracture energy degradation due to the repeated externally applied loads is introduced as a function of a local energy accumulation variable, which takes the structural loading history into account. To this end, a novel definition of the energy accumulation variable is proposed, allowing the fracture analysis at monotonic loading without the interference of the fatigue extension, thus making the framework generalised. Moreover, this definition includes the mean load influence of implicitly. The elastoplastic material model with the combined nonlinear isotropic and nonlinear kinematic hardening is introduced to account for cyclic plasticity. The ability of the proposed phenomenological approach to naturally recover main features of fatigue, including Paris law and Wöhler curve under different load ratios is presented through numerical examples and compared with experimental data from the author’s previous work. Physical interpretation of additional fatigue material parameter is explored through the parametric study. © 2021, The Author(s)
eXtended Variational Quasicontinuum Methodology for Lattice Networks with Damage and Crack Propagation
Lattice networks with dissipative interactions are often employed to analyze
materials with discrete micro- or meso-structures, or for a description of
heterogeneous materials which can be modelled discretely. They are, however,
computationally prohibitive for engineering-scale applications. The
(variational) QuasiContinuum (QC) method is a concurrent multiscale approach
that reduces their computational cost by fully resolving the (dissipative)
lattice network in small regions of interest while coarsening elsewhere. When
applied to damageable lattices, moving crack tips can be captured by adaptive
mesh refinement schemes, whereas fully-resolved trails in crack wakes can be
removed by mesh coarsening. In order to address crack propagation efficiently
and accurately, we develop in this contribution the necessary generalizations
of the variational QC methodology. First, a suitable definition of crack paths
in discrete systems is introduced, which allows for their geometrical
representation in terms of the signed distance function. Second, special
function enrichments based on the partition of unity concept are adopted, in
order to capture kinematics in the wakes of crack tips. Third, a summation rule
that reflects the adopted enrichment functions with sufficient degree of
accuracy is developed. Finally, as our standpoint is variational, we discuss
implications of the mesh refinement and coarsening from an energy-consistency
point of view. All theoretical considerations are demonstrated using two
numerical examples for which the resulting reaction forces, energy evolutions,
and crack paths are compared to those of the direct numerical simulations.Comment: 36 pages, 23 figures, 1 table, 2 algorithms; small changes after
review, paper title change
Resurgence in Deformed Integrable Models
Resurgence has been shown to be a powerful and even necessary technique to understand many physical system. The study of perturbative methods in general quantum field theories is hard, but progress is often possible in reduced settings, such as integrable models. In this thesis, we study resurgent effects in integrable deformations of two-dimensional σ-models in two settings.First, we study the integrable bi-Yang-Baxter deformation of the SU(2) principal chiral model (PCM) and find finite action uniton and complex uniton solutions. Under an adiabatic compactification on an S1, we obtain a quantum mechanical system with an elliptic Lam´e-like potential. We perform a perturbative calculation of the ground state energy of this quantum mechanical system to large orders obtaining an asymptotic series. Using the Borel-Pad´e technique, we determine that the locations of branch cuts in the Borel plane match the values of the uniton and complex uniton actions. Therefore, we can match the non-perturbative contributions to the energy with the uniton solutionswhich fractionate upon adiabatic compactification. An off-shoot of the WKB analysis, is to identify the quadratic differential of this deformed PCM with that of an N = 2 Seiberg-Witten theory. This can be done either as an Nf = 4 SU(2) theory or as an elliptic SU(2) × SU(2) quiver theory. The mass parameters of the gauge theory are given in terms of the bi-Yang-Baxter deformation parameters.Second, we perform a perturbative expansion of the thermodynamic Bethe ansatz (TBA) equations of the SU(N) λ-model with WZW level k in the presence of a chemical potential. This is done with its exact S-matrix and the recently developed techniques [1, 2] using a Wiener-Hopf decomposition, which involve a careful matching of bulk and edge ans¨atze. We determine the asymptotic expansion of this series and compute its renormalon ambiguities in the Borel plane. The analysis is supplemented by a parallel solution of the TBA equations that results in a transseries. The transseries comes with an ambiguity that is shown to precisely match the Borel ambiguity. It is shown that the leading IR renormalon vanishes when k is a divisor of N
Optimisation of manufacturing process parameters for variable component geometries using reinforcement learning
Tailoring manufacturing processes to optimum part quality often requires numerous resource-intensive trial experiments in practice. Physics-based process simulations in combination with general-purpose optimisation algorithms allow for an a priori process optimisation and help concentrate costly trials on the most promising variants. However, considerable computation times are a significant barrier, especially for iterative optimisation. Surrogate-based optimisation often helps reduce the computational effort but surrogate models are typically case-specific and cannot adapt to different manufacturing situations. Consequently, even minor problem variations e.g. geometry adaptions invalidate the surrogate and require resampling of data and retraining of the surrogate. Reinforcement Learning aims at inferring optimal actions in variable situations. In this work, it is used to train a neural network to estimate optimal process parameters (“actions”) for variable component geometries (“situations”). The use case is fabric forming in which pressure pads are positioned to optimise the material intake. After training, the network is found to give meaningful parameter estimations even for new geometries not considered during training. Thus, it extracts reusable information from generic process samples and successfully applies it to new, non-generic components. Since data is reused rather than resampled, the approach is deemed a promising option for lean part and process development
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