10,381 research outputs found
Beam scanning by liquid-crystal biasing in a modified SIW structure
A fixed-frequency beam-scanning 1D antenna based on Liquid Crystals (LCs) is designed for application in 2D scanning with lateral alignment. The 2D array environment imposes full decoupling of adjacent 1D antennas, which often conflicts with the LC requirement of DC biasing: the proposed design accommodates both. The LC medium is placed inside a Substrate Integrated Waveguide (SIW) modified to work as a Groove Gap Waveguide, with radiating slots etched on the upper broad wall, that radiates as a Leaky-Wave Antenna (LWA). This allows effective application of the DC bias voltage needed for tuning the LCs. At the same time, the RF field remains laterally confined, enabling the possibility to lay several antennas in parallel and achieve 2D beam scanning. The design is validated by simulation employing the actual properties of a commercial LC medium
Unstable Periodic Orbits: a language to interpret the complexity of chaotic systems
Unstable periodic orbits (UPOs), exact periodic solutions of the evolution equation, offer a very
powerful framework for studying chaotic dynamical systems, as they allow one to dissect their
dynamical structure. UPOs can be considered the skeleton of chaotic dynamics, its essential
building blocks. In fact, it is possible to prove that in a chaotic system, UPOs are dense in
the attractor, meaning that it is always possible to find a UPO arbitrarily near any chaotic
trajectory. We can thus think of the chaotic trajectory as being approximated by different
UPOs as it evolves in time, jumping from one UPO to another as a result of their instability.
In this thesis we provide a contribution towards the use of UPOs as a tool to understand and
distill the dynamical structure of chaotic dynamical systems. We will focus on two models,
characterised by different properties, the Lorenz-63 and Lorenz-96 model.
The process of approximation of a chaotic trajectory in terms of UPOs will play a central role
in our investigation. In fact, we will use this tool to explore the properties of the attractor of
the system under the lens of its UPOs.
In the first part of the thesis we consider the Lorenz-63 model with the classic parameters’ value.
We investigate how a chaotic trajectory can be approximated using a complete set of UPOs
up to symbolic dynamics’ period 14. At each instant in time, we rank the UPOs according to
their proximity to the position of the orbit in the phase space. We study this process from
two different perspectives. First, we find that longer period UPOs overwhelmingly provide the
best local approximation to the trajectory. Second, we construct a finite-state Markov chain
by studying the scattering of the trajectory between the neighbourhood of the various UPOs.
Each UPO and its neighbourhood are taken as a possible state of the system. Through the
analysis of the subdominant eigenvectors of the corresponding stochastic matrix we provide a
different interpretation of the mixing processes occurring in the system by taking advantage of
the concept of quasi-invariant sets.
In the second part of the thesis we provide an extensive numerical investigation of the variability
of the dynamical properties across the attractor of the much studied Lorenz ’96 dynamical
system. By combining the Lyapunov analysis of the tangent space with the study of the
shadowing of the chaotic trajectory performed by a very large set of unstable periodic orbits,
we show that the observed variability in the number of unstable dimensions, which shows a
serious breakdown of hyperbolicity, is associated with the presence of a substantial number of
finite-time Lyapunov exponents that fluctuate about zero also when very long averaging times
are considered
An approximate maximum likelihood estimator of drift parameters in a multidimensional diffusion model
For a fixed and , a -dimensional vector stochastic
differential equation is studied over a
time interval . Vector of drift parameters is unknown. The
dependence in is in general nonlinear. We prove that the difference
between approximate maximum likelihood estimator of the drift parameter
obtained from discrete
observations and maximum likelihood
estimator obtained from continuous
observations , when tends to zero,
converges stably in law to the mixed normal random vector with covariance
matrix that depends on and on path . The
uniform ellipticity of diffusion matrix emerges as the
main assumption on the diffusion coefficient function.Comment: 38 page
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Fourth Order Dispersion in Nonlinear Media
In recent years, there has been an explosion of interest in media bearing quarticdispersion. After the experimental realization of so-called pure-quartic solitons, asignificant number of studies followed both for bright and for dark solitonic struc-tures exploring the properties of not only quartic, but also setic, octic, decic etc.dispersion, but also examining the competition between, e.g., quadratic and quarticdispersion among others.In the first chapter of this Thesis, we consider the interaction of solitary waves ina model involving the well-known φ4 Klein-Gordon theory, bearing both Laplacian and biharmonic terms with different prefactors. As a result of the competition ofthe respective linear operators, we obtain three distinct cases as we vary the modelparameters. In the first the biharmonic effect dominates, yielding an oscillatoryinter-wave interaction; in the third the harmonic effect prevails yielding exponen-tial interactions, while we find an intriguing linearly modulated exponential effectin the critical second case, separating the above two regimes. For each case, wecalculate the force between the kink and antikink when initially separated with suf-ficient distance. Being able to write the acceleration as a function of the separationdistance, and its corresponding ordinary differential equation, we test the corre-sponding predictions, finding very good agreement, where appropriate, with thecorresponding partial differential equation results. Where the two findings differ,we explain the source of disparities. Finally, we offer a first glimpse of the interplayof harmonic and biharmonic effects on the results of kink-antikink collisions andthe corresponding single- and multi-bounce windows.In the next two Chapters, we explore the competition of quadratic and quar-tic dispersion in producing kink-like solitary waves in a model of the nonlinearSchroedinger type bearing cubic nonlinearity. We present 6 families of multikink so-lutions and explore their bifurcations as a prototypical parameter is varied, namelythe strength of the quadratic dispersion. We reveal a rich bifurcation structure forthe system, connecting two-kink states with ones involving 4-, as well as 6-kinks.The stability of all of these states is explored. For each family, we discuss a “lowerbranch” adhering to the energy landscape of the 2-kink states (also discussed inthe previous Chapter). We also, however, study in detail the “upper branches”bearing higher numbers of kinks. In addition to computing the stationary statesand analyzing their stability at the PDE level, we develop an effective particle the-ory that is shown to be surprisingly efficient in capturing the kink equilibria and normal (as well as unstable) modes. Finally, the results of the bifurcation analysisare corroborated with direct numerical simulations involving the excitation of thestates in a targeted way in order to explore their instability-induced dynamics.While the previous two studies were focused on the one-dimensional problem,in the fourth and final chapter, we explore a two-dimensional realm. More specif-ically, we provide a characterization of the ground states of a higher-dimensionalquadratic-quartic model of the nonlinear Schr ̈odinger class with a combination of afocusing biharmonic operator with either an isotropic or an anisotropic defocusingLaplacian operator (at the linear level) and power-law nonlinearity. Examiningprincipally the prototypical example of dimension d = 2, we find that instabilityarises beyond a certain threshold coefficient of the Laplacian between the cubic andquintic cases, while all solutions are stable for powers below the cubic. Above thequintic, and up to a critical nonlinearity exponent p, there exists a progressivelynarrowing range of stable frequencies. Finally, above the critical p all solutionsare unstable. The picture is rather similar in the anisotropic case, with the dif-ference that even before the cubic case, the numerical computations suggest aninterval of unstable frequencies. Our analysis generalizes the relevant observationsfor arbitrary combinations of Laplacian prefactor b and nonlinearity power p.We conclude the thesis with a summary of its main findings, as well as with anoutlook towards interesting future problem
Linear and Nonlinear Kinetic Alfv\'en Wave Physics in Cylindrical Plasmas
Kinetic Alfv\'en Waves (KAWs) are generated in magnetized space and
laboratory plasmas due to a continuous shear Alfv\'en wave (SAW) spectrum and,
unlike SAWs, are characterized by microscale perpendicular structures of the
order of the thermal ion Larmor radius. This has important consequences on
heating, acceleration and transport processes connected with KAWs.
Historically, KAWs generation by mode conversion of SAWs in laboratory plasmas
and their strong damping/absorption right after SAW mode conversion have been
investigated for plasma heating. Here, we focus on the opposite limit: a mode
converted KAW weakly absorbed in a periodic magnetized plasma cylinder. We show
that a KAW may be excited as resonant cavity mode in the region between the
magnetic axis and the SAW resonant layer generated externally by an antenna
launcher; this process is qualitatively similar to mode converted electron
Bernstein waves. In this way, large amplitude KAWs may be generated time
asymptotically with relatively small coupled antenna power. This case has
little or no relevance for plasma heating but interesting nonlinear
implications for plasma equilibrium. In particular, we demonstrate that KAWs
may generate convective cells (CCs) by modulational instability, that a
consequence of plasma nonuniformity is the azimuthal symmetry breaking due to
plasma diamagnetic effects, that the modulational instability growth rate is
enhanced over the corresponding uniform plasma limit, that the unstable
parameter space is extended, and that the cylindrical geometry causes a complex
interplay between nonlinearity and nonuniformity. As a result, we show that it
is possible to control the CC radial structures and the corresponding parallel
electric field generation not only by means of the antenna frequency but also
by fine tuning of its amplitude.Comment: 117 pages, 42 figure
Modal sensitivity of three-dimensional planetary geared rotor systems to planet gear parameters
A parameter study is presented to determine effects of planet gear design parameters on the global modal behaviour of planetary geared rotor systems. The modal sensitivity analysis is conducted using a three-dimensional dynamic model of a planetary geared rotor system for the number of planet gears, planet mistuning, mass of planet gears, gear mesh stiffness and planet gear speed. These parameters have varying impacts on both natural frequencies and mode shapes, therefore the sensitivity of the planetary geared rotor vibration modes to the planet gear parameters is determined by computing the frequency shifts and comparing the mode shapes. The results show that the mass and mesh stiffness of planet gears have a larger influence on the global dynamic response. Torsional modes and coupled torsional-axial modes are more sensitive to gear mesh stiffness whereas lateral vibration modes are more sensitive to gearbox mass. Planet mistuning results in coupling between lateral and torsional vibrations. The planetary gearbox becomes more rigid in the torsional-axial modes and more flexible in the lateral modes with an increase in the number of planet gears. Planet gears are also found to be having significant gyroscopic effects inside the planetary gearbox. The main original findings in this study can be directly used as initial guidelines for planetary geared rotor design
Soliton Gas: Theory, Numerics and Experiments
The concept of soliton gas was introduced in 1971 by V. Zakharov as an
infinite collection of weakly interacting solitons in the framework of
Korteweg-de Vries (KdV) equation. In this theoretical construction of a diluted
soliton gas, solitons with random parameters are almost non-overlapping. More
recently, the concept has been extended to dense gases in which solitons
strongly and continuously interact. The notion of soliton gas is inherently
associated with integrable wave systems described by nonlinear partial
differential equations like the KdV equation or the one-dimensional nonlinear
Schr\"odinger equation that can be solved using the inverse scattering
transform. Over the last few years, the field of soliton gases has received a
rapidly growing interest from both the theoretical and experimental points of
view. In particular, it has been realized that the soliton gas dynamics
underlies some fundamental nonlinear wave phenomena such as spontaneous
modulation instability and the formation of rogue waves. The recently
discovered deep connections of soliton gas theory with generalized
hydrodynamics have broadened the field and opened new fundamental questions
related to the soliton gas statistics and thermodynamics. We review the main
recent theoretical and experimental results in the field of soliton gas. The
key conceptual tools of the field, such as the inverse scattering transform,
the thermodynamic limit of finite-gap potentials and the Generalized Gibbs
Ensembles are introduced and various open questions and future challenges are
discussed.Comment: 35 pages, 8 figure
Structured Dynamic Pricing: Optimal Regret in a Global Shrinkage Model
We consider dynamic pricing strategies in a streamed longitudinal data set-up
where the objective is to maximize, over time, the cumulative profit across a
large number of customer segments. We consider a dynamic probit model with the
consumers' preferences as well as price sensitivity varying over time. Building
on the well-known finding that consumers sharing similar characteristics act in
similar ways, we consider a global shrinkage structure, which assumes that the
consumers' preferences across the different segments can be well approximated
by a spatial autoregressive (SAR) model. In such a streamed longitudinal
set-up, we measure the performance of a dynamic pricing policy via regret,
which is the expected revenue loss compared to a clairvoyant that knows the
sequence of model parameters in advance. We propose a pricing policy based on
penalized stochastic gradient descent (PSGD) and explicitly characterize its
regret as functions of time, the temporal variability in the model parameters
as well as the strength of the auto-correlation network structure spanning the
varied customer segments. Our regret analysis results not only demonstrate
asymptotic optimality of the proposed policy but also show that for policy
planning it is essential to incorporate available structural information as
policies based on unshrunken models are highly sub-optimal in the
aforementioned set-up.Comment: 34 pages, 5 figure
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
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
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