10,926 research outputs found
Quantum resonances and analysis of the survival amplitude in the nonlinear Winter's model
In this paper we show that the typical effects of quantum resonances, namely,
the exponential-type decay of the survival amplitude, continue to exist even
when a nonlinear perturbative term is added to the time-dependent Schroedinger
equation. The difficulty in giving a rigorous and appropriate definition of
quantum resonances by means of the notions already used for linear equations is
also highlighted.Comment: 31 pages, 8 figure
Optimal Control of the Landau-de Gennes Model of Nematic Liquid Crystals
We present an analysis and numerical study of an optimal control problem for
the Landau-de Gennes (LdG) model of nematic liquid crystals (LCs), which is a
crucial component in modern technology. They exhibit long range orientational
order in their nematic phase, which is represented by a tensor-valued (spatial)
order parameter . Equilibrium LC states correspond to functions
that (locally) minimize an LdG energy functional. Thus, we consider an
-gradient flow of the LdG energy that allows for finding local minimizers
and leads to a semi-linear parabolic PDE, for which we develop an optimal
control framework. We then derive several a priori estimates for the forward
problem, including continuity in space-time, that allow us to prove existence
of optimal boundary and external ``force'' controls and to derive optimality
conditions through the use of an adjoint equation. Next, we present a simple
finite element scheme for the LdG model and a straightforward optimization
algorithm. We illustrate optimization of LC states through numerical
experiments in two and three dimensions that seek to place LC defects (where
) in desired locations, which is desirable in applications.Comment: 26 pages, 9 figure
Worldtube excision method for intermediate-mass-ratio inspirals: scalar-field model in 3+1 dimensions
Binary black hole simulations become increasingly more computationally
expensive with smaller mass ratios, partly because of the longer evolution
time, and partly because the lengthscale disparity dictates smaller time steps.
The program initiated by Dhesi et al. (arXiv:2109.03531) explores a method for
alleviating the scale disparity in simulations with mass ratios in the
intermediate astrophysical range (), where
purely perturbative methods may not be adequate. A region ("worldtube") much
larger than the small black hole is excised from the numerical domain, and
replaced with an analytical model approximating a tidally deformed black hole.
Here we apply this idea to a toy model of a scalar charge in a fixed circular
geodesic orbit around a Schwarzschild black hole, solving for the massless
Klein-Gordon field. This is a first implementation of the worldtube excision
method in full 3+1 dimensions. We demonstrate the accuracy and efficiency of
the method, and discuss the steps towards applying it for evolving orbits and,
ultimately, in the binary black-hole scenario. Our implementation is publicly
accessible in the SpECTRE numerical relativity code.Comment: 19 pages, 10 figure
Projected Multi-Agent Consensus Equilibrium (PMACE) for Distributed Reconstruction with Application to Ptychography
Multi-Agent Consensus Equilibrium (MACE) formulates an inverse imaging
problem as a balance among multiple update agents such as data-fitting terms
and denoisers. However, each such agent operates on a separate copy of the full
image, leading to redundant memory use and slow convergence when each agent
affects only a small subset of the full image. In this paper, we extend MACE to
Projected Multi-Agent Consensus Equilibrium (PMACE), in which each agent
updates only a projected component of the full image, thus greatly reducing
memory use for some applications.We describe PMACE in terms of an equilibrium
problem and an equivalent fixed point problem and show that in most cases the
PMACE equilibrium is not the solution of an optimization problem. To
demonstrate the value of PMACE, we apply it to the problem of ptychography, in
which a sample is reconstructed from the diffraction patterns resulting from
coherent X-ray illumination at multiple overlapping spots. In our PMACE
formulation, each spot corresponds to a separate data-fitting agent, with the
final solution found as an equilibrium among all the agents. Our results
demonstrate that the PMACE reconstruction algorithm generates more accurate
reconstructions at a lower computational cost than existing ptychography
algorithms when the spots are sparsely sampled
Stability of space-time isogeometric methods for wave propagation problems
This thesis aims at investigating the first steps toward an unconditionally
stable space-time isogeometric method, based on splines of maximal regularity,
for the linear acoustic wave equation. The unconditional stability of
space-time discretizations for wave propagation problems is a topic of
significant interest, by virtue of the advantages of space-time methods
compared with more standard time-stepping techniques. In the case of continuous
finite element methods, several stabilizations have been proposed. Inspired by
one of these works, we address the stability issue by studying the isogeometric
method for an ordinary differential equation closely related to the wave
equation. As a result, we provide a stabilized isogeometric method whose
effectiveness is supported by numerical tests. Motivated by these results, we
conclude by suggesting an extension of this stabilization tool to the
space-time isogeometric formulation of the acoustic wave equation.Comment: Masters thesi
Examples of works to practice staccato technique in clarinet instrument
Klarnetin staccato tekniğini güçlendirme aşamaları eser çalışmalarıyla uygulanmıştır. Staccato
geçişlerini hızlandıracak ritim ve nüans çalışmalarına yer verilmiştir. Çalışmanın en önemli amacı
sadece staccato çalışması değil parmak-dilin eş zamanlı uyumunun hassasiyeti üzerinde de
durulmasıdır. Staccato çalışmalarını daha verimli hale getirmek için eser çalışmasının içinde etüt
çalışmasına da yer verilmiştir. Çalışmaların üzerinde titizlikle durulması staccato çalışmasının ilham
verici etkisi ile müzikal kimliğe yeni bir boyut kazandırmıştır. Sekiz özgün eser çalışmasının her
aşaması anlatılmıştır. Her aşamanın bir sonraki performans ve tekniği güçlendirmesi esas alınmıştır.
Bu çalışmada staccato tekniğinin hangi alanlarda kullanıldığı, nasıl sonuçlar elde edildiği bilgisine
yer verilmiştir. Notaların parmak ve dil uyumu ile nasıl şekilleneceği ve nasıl bir çalışma disiplini
içinde gerçekleşeceği planlanmıştır. Kamış-nota-diyafram-parmak-dil-nüans ve disiplin
kavramlarının staccato tekniğinde ayrılmaz bir bütün olduğu saptanmıştır. Araştırmada literatür
taraması yapılarak staccato ile ilgili çalışmalar taranmıştır. Tarama sonucunda klarnet tekniğin de
kullanılan staccato eser çalışmasının az olduğu tespit edilmiştir. Metot taramasında da etüt
çalışmasının daha çok olduğu saptanmıştır. Böylelikle klarnetin staccato tekniğini hızlandırma ve
güçlendirme çalışmaları sunulmuştur. Staccato etüt çalışmaları yapılırken, araya eser çalışmasının
girmesi beyni rahatlattığı ve istekliliği daha arttırdığı gözlemlenmiştir. Staccato çalışmasını yaparken
doğru bir kamış seçimi üzerinde de durulmuştur. Staccato tekniğini doğru çalışmak için doğru bir
kamışın dil hızını arttırdığı saptanmıştır. Doğru bir kamış seçimi kamıştan rahat ses çıkmasına
bağlıdır. Kamış, dil atma gücünü vermiyorsa daha doğru bir kamış seçiminin yapılması gerekliliği
vurgulanmıştır. Staccato çalışmalarında baştan sona bir eseri yorumlamak zor olabilir. Bu açıdan
çalışma, verilen müzikal nüanslara uymanın, dil atış performansını rahatlattığını ortaya koymuştur.
Gelecek nesillere edinilen bilgi ve birikimlerin aktarılması ve geliştirici olması teşvik edilmiştir.
Çıkacak eserlerin nasıl çözüleceği, staccato tekniğinin nasıl üstesinden gelinebileceği anlatılmıştır.
Staccato tekniğinin daha kısa sürede çözüme kavuşturulması amaç edinilmiştir. Parmakların
yerlerini öğrettiğimiz kadar belleğimize de çalışmaların kaydedilmesi önemlidir. Gösterilen azmin ve
sabrın sonucu olarak ortaya çıkan yapıt başarıyı daha da yukarı seviyelere çıkaracaktır
Random Young towers and quenched decay of correlations for predominantly expanding multimodal circle maps
In this paper, we study the random dynamical system generated by
a family of maps $f_{\omega_0}(x) = \alpha \xi (x+\omega_0) +a\
(\mathrm{mod }\ 1),\xi: \mathbb S^1 \to \mathbb Ra\in \mathbb S^1\alpha,\varepsilon>0c\in (0,1)\alpha\varepsilon > \alpha^{-1+c},f_\omega^n$
presents a random Young tower structure and quenched decay of correlations.Comment: 38 pages, 0 figure
Optimal distributed control for a viscous non-local tumour growth model
In this paper, we address an optimal distributed control problem for a
non-local model of phase-field type, describing the evolution of tumour cells
in presence of a nutrient. The model couples a non-local and viscous
Cahn-Hilliard equation for the phase parameter with a reaction-diffusion
equation for the nutrient. The optimal control problem aims at finding a
therapy, encoded as a source term in the system, both in the form of
radiotherapy and chemotherapy, which could lead to the evolution of the phase
variable towards a desired final target. First, we prove strong well-posedness
for the system of non-linear partial differential equations. In particular, due
to the presence of a viscous regularisation, we can also consider double-well
potentials of singular type and cross-diffusion terms related to the effects of
chemotaxis. Moreover, the particular structure of the reaction terms allows us
to prove new regularity results for this kind of system. Then, turning to the
optimal control problem, we prove the existence of an optimal therapy and, by
studying Fr\'echet-differentiability properties of the control-to-state
operator and the corresponding adjoint system, we obtain the first-order
necessary optimality conditions.Comment: 43 page
Limit theorems for non-Markovian and fractional processes
This thesis examines various non-Markovian and fractional processes---rough volatility models, stochastic Volterra equations, Wiener chaos expansions---through the prism of asymptotic analysis.
Stochastic Volterra systems serve as a conducive framework encompassing most rough volatility models used in mathematical finance. In Chapter 2, we provide a unified treatment of pathwise large and moderate deviations principles for a general class of multidimensional stochastic Volterra equations with singular kernels, not necessarily of convolution form. Our methodology is based on the weak convergence approach by Budhiraja, Dupuis and Ellis.
This powerful approach also enables us to investigate the pathwise large deviations of families of white noise functionals characterised by their Wiener chaos expansion as~
In Chapter 3, we provide sufficient conditions for the large deviations principle to hold in path space, thereby refreshing a problem left open By Pérez-Abreu (1993). Hinging on analysis on Wiener space, the proof involves describing, controlling and identifying the limit of perturbed multiple stochastic integrals.
In Chapter 4, we come back to mathematical finance via the route of Malliavin calculus. We present explicit small-time formulae for the at-the-money implied volatility, skew and curvature in a large class of models, including rough volatility models and their multi-factor versions. Our general setup encompasses both European options on a stock and VIX options. In particular, we develop a detailed analysis of the two-factor rough Bergomi model.
Finally, in Chapter 5, we consider the large-time behaviour of affine stochastic Volterra equations, an under-developed area in the absence of Markovianity.
We leverage on a measure-valued Markovian lift introduced by Cuchiero and Teichmann and the associated notion of generalised Feller property.
This setting allows us to prove the existence of an invariant measure for the lift and hence of a stationary distribution for the affine Volterra process, featuring in the rough Heston model.Open Acces
Sparse Functional Linear Discriminant Analysis
Functional linear discriminant analysis offers a simple yet efficient method for classification, with the possibility of achieving a perfect classification. Several methods are proposed in the literature that mostly address the dimensionality of the problem. On the other hand, there is a growing interest in interpretability of the analysis, which favors a simple and sparse solution. In this work, we propose a new approach that incorporates a type of sparsity that identifies nonzero sub-domains in the functional setting, offering a solution that is easier to interpret without compromising performance. With the need to embed additional constraints in the solution, we reformulate the functional linear discriminant analysis as a regularization problem with an appropriate penalty. Inspired by the success of ℓ1-type regularization at inducing zero coefficients for scalar variables, we develop a new regularization method for functional linear discriminant analysis that incorporates an L1-type penalty, ∫ |f|, to induce zero regions. We demonstrate that our formulation has a well-defined solution that contains zero regions, achieving a functional sparsity in the sense of domain selection. In addition, the misclassification probability of the regularized solution is shown to converge to the Bayes error if the data are Gaussian. Our method does not presume that the underlying function has zero regions in the domain, but produces a sparse estimator that consistently estimates the true function whether or not the latter is sparse. Numerical comparisons with existing methods demonstrate this property in finite samples with both simulated and real data examples
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