34,920 research outputs found
Fractal functions on the real projective plane
Formerly the geometry was based on shapes, but since the last centuries this
founding mathematical science deals with transformations, projections and
mappings. Projective geometry identifies a line with a single point, like the
perspective on the horizon line and, due to this fact, it requires a
restructuring of the real mathematical and numerical analysis. In particular,
the problem of interpolating data must be refocused. In this paper we define a
linear structure along with a metric on a projective space, and prove that the
space thus constructed is complete. Then we consider an iterated function
system giving rise to a fractal interpolation function of a set of data.Comment: 25 pages, 18 figure
Decentralized projected Riemannian gradient method for smooth optimization on compact submanifolds
We consider the problem of decentralized nonconvex optimization over a
compact submanifold, where each local agent's objective function defined by the
local dataset is smooth. Leveraging the powerful tool of proximal smoothness,
we establish local linear convergence of the projected gradient descent method
with unit step size for solving the consensus problem over the compact
manifold. This serves as the basis for analyzing decentralized algorithms on
manifolds. Then, we propose two decentralized methods, namely the decentralized
projected Riemannian gradient descent (DPRGD) and the decentralized projected
Riemannian gradient tracking (DPRGT) methods. We establish their convergence
rates of and , respectively, to
reach a stationary point. To the best of our knowledge, DPRGT is the first
decentralized algorithm to achieve exact convergence for solving decentralized
optimization over a compact manifold. The key ingredients in the proof are the
Lipschitz-type inequalities of the projection operator on the compact manifold
and smooth functions on the manifold, which could be of independent interest.
Finally, we demonstrate the effectiveness of our proposed methods compared to
state-of-the-art ones through numerical experiments on eigenvalue problems and
low-rank matrix completion.Comment: 32 page
Quantum Mechanics Lecture Notes. Selected Chapters
These are extended lecture notes of the quantum mechanics course which I am
teaching in the Weizmann Institute of Science graduate physics program. They
cover the topics listed below. The first four chapter are posted here. Their
content is detailed on the next page. The other chapters are planned to be
added in the coming months.
1. Motion in External Electromagnetic Field. Gauge Fields in Quantum
Mechanics.
2. Quantum Mechanics of Electromagnetic Field
3. Photon-Matter Interactions
4. Quantization of the Schr\"odinger Field (The Second Quantization)
5. Open Systems. Density Matrix
6. Adiabatic Theory. The Berry Phase. The Born-Oppenheimer Approximation
7. Mean Field Approaches for Many Body Systems -- Fermions and Boson
An iterative warping and clustering algorithm to estimate multiple wave-shape functions from a nonstationary oscillatory signal
Nonsinusoidal oscillatory signals are everywhere. In practice, the
nonsinusoidal oscillatory pattern, modeled as a 1-periodic wave-shape function
(WSF), might vary from cycle to cycle. When there are finite different WSFs,
, so that the WSF jumps from one to another suddenly, the
different WSFs and jumps encode useful information. We present an iterative
warping and clustering algorithm to estimate from a
nonstationary oscillatory signal with time-varying amplitude and frequency, and
hence the change points of the WSFs. The algorithm is a novel combination of
time-frequency analysis, singular value decomposition entropy and vector
spectral clustering. We demonstrate the efficiency of the proposed algorithm
with simulated and real signals, including the voice signal, arterial blood
pressure, electrocardiogram and accelerometer signal. Moreover, we provide a
mathematical justification of the algorithm under the assumption that the
amplitude and frequency of the signal are slowly time-varying and there are
finite change points that model sudden changes from one wave-shape function to
another one.Comment: 39 pages, 11 figure
A hybrid quantum algorithm to detect conical intersections
Conical intersections are topologically protected crossings between the
potential energy surfaces of a molecular Hamiltonian, known to play an
important role in chemical processes such as photoisomerization and
non-radiative relaxation. They are characterized by a non-zero Berry phase,
which is a topological invariant defined on a closed path in atomic coordinate
space, taking the value when the path encircles the intersection
manifold. In this work, we show that for real molecular Hamiltonians, the Berry
phase can be obtained by tracing a local optimum of a variational ansatz along
the chosen path and estimating the overlap between the initial and final state
with a control-free Hadamard test. Moreover, by discretizing the path into
points, we can use single Newton-Raphson steps to update our state
non-variationally. Finally, since the Berry phase can only take two discrete
values (0 or ), our procedure succeeds even for a cumulative error bounded
by a constant; this allows us to bound the total sampling cost and to readily
verify the success of the procedure. We demonstrate numerically the application
of our algorithm on small toy models of the formaldimine molecule
(\ce{H2C=NH}).Comment: 15 + 10 pages, 4 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
Airportscape and its Effect on Airport Sense of Place and Destination Image Perception
Purpose – This study aims to validate a conceptual definition of airportscape and develop a multidimensional scale that integrates servicescape and service quality dimensions in order to comprehensively investigate airport service management. In addition, the study examines the structural relationships amongst airportscape, sense of place, airport image and destination image.
Design/methodology/approach – Covariance-based structural equation modelling has been employed. This study has collected the responses from 1,189 Thai respondents who had their experience in an international airport in the past 12 months.
Findings – Key findings reveal the set of three airportscape attributes which positively influenced the air traveller’s perceived sense of place. Four other dimensions were found to positively influence the airport image. The results also suggested the positive relationships amongst sense of place, airport image and destination image. Sense of place strongly predicted the destination image and airport image, and was found to be an important mediator of the relationship between airportscape dimension and perceived image variables
Originality/value – The study validates the airportscape scale and introduces “sense of place”, a concept that has not been objectively investigated in the airport context, and in relation to tourism. The findings provide insights to airport managers and tourism authorities by examining areas that highlight an airport’s sense of place and representation of the destination. The study also strengthens the theoretical link between airport and tourism knowledge, by showing that the airport’s sense of place can strongly influence airport image and destination image. The result ascertains that the airport can be a representative of a destination through the creation of a sense of place
Metaphors of London fog, smoke and mist in Victorian and Edwardian Art and Literature
Julian Wolfreys has argued that after 1850 writers employed stock images of the city without allowing them to transform their texts. This thesis argues, on the contrary, that metaphorical uses of London fog were complex and subtle during the Victorian and Edwardian periods, at least until 1914. Fog represented, in particular, formlessness and the dissolution of boundaries. Examining the idea of fog in literature, verse, newspaper accounts and journal articles, as well as in the visual arts, as part of a common discourse about London and the state of its inhabitants, this thesis charts how the metaphorical appropriation of this idea changed over time. Four of Dickens's novels are used to track his use of fog as part of a discourse of the natural and unnatural in individual and society, identifying it with London in progressively more negative terms. Visual representations of fog by Constable, Turner, Whistler, Monet, Markino, O'Connor, Roberts and Wyllie and Coburn showed an increasing readiness to engage with this discourse. Social tensions in the city in the 1880s were articulated in art as well as in fiction. Authors like Hay and Barr showed the destruction of London by its fog because of its inhabitants' supposed degeneracy. As the social threat receded, apocalyptic scenarios gave way to a more optimistic view in the work of Owen and others. Henry James used fog as a metaphorical representation of the boundaries of gendered behaviour in public, and the problems faced by women who crossed them. The dissertation also examines fog and individual transgression, in novels and short stories by Lowndes, Stevenson, Conan Doyle and Joseph Conrad. After 1914, fog was no more than a crude signifier of Victorian London in literature, film and, later, television, deployed as a cliche instead of the subtle metaphorical idea discussed in this thesis
The temporality of rhetoric: the spatialization of time in modern criticism
Every conception of criticism conceals a notion of time which informs the manner in which the critic conceives of history, representation and criticism itself. This thesis reveals the philosophies of time inherent in certain key modern critical concepts: allegory, irony and the sublime. Each concept opens a breach in time, a disruption of chronology. In each case this gap or aporia is emphatically closed, elided or denied. Taking the philosophy of time elaborated by Giorgio Agamben as an introductory proposition, my argument turns in Chapter One to the allegorical temporality which Walter Benjamin sees as the time of photography. The second chapter examines the aesthetics of the sublime as melancholic or mournful untimeliness. In Chapter Three, Paul de Man's conception of irony provides an exemplary instance of the denial of this troubling temporal predicament. In opposition to the foreclosure of the disturbing temporalities of criticism, history and representation, the thesis proposes a fundamental rethinking of the philosophy of time as it relates to these categories of reflection. In a reading of an inaugural meditation on the nature of time, and in examining certain key contemporary philosophical and critical texts, I argue for a critical attendance to that which eludes those modes of thought that attempt to map time as a recognizable and essentially spatial field. The Confessions of Augustine provide, in the fourth chapter, a model for thinking through the problems set up earlier: Augustine affords us, precisely, a means of conceiving of the gap or the interim. In the final chapter, this concept is developed with reference to the criticism of Arnold and Eliot, the fiction of Virginia Woolf and the philosophy of cinema derived from Deleuze and Lyotard. In conclusion, the philosophical implications of the thesis are placed in relation to a conception of the untimeliness of death
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