162,383 research outputs found
Fermionic quantization of Hopf solitons
No full textIn this paper we show how to quantize Hopf solitons using the Finkelstein-Rubinstein approach. Hopf solitons can be quantized as fermions if their Hopf charge is odd. Symmetries of classical minimal energy configurations induce loops in configuration space which give rise to constraints on the wave function. These constraints depend on whether the given loop is contractible. Our method is to exploit the relationship between the configuration spaces of the Faddeev-Hopf and Skyrme models provided by the Hopf fibration. We then use recent results in the Skyrme model to determine whether loops are contractible. We discuss possible quantum ground states up to Hopf charge Q=7
Regularidade de operadores de Wiener-Hopf mais Hankel
No full textDoutoramento em MatemáticaIn this thesis we consider Wiener-Hopf-Hankel operators with Fourier symbols
in the class of almost periodic, semi-almost periodic and piecewise almost
periodic functions. In the first place, we consider Wiener-Hopf-Hankel
operators acting between L2 Lebesgue spaces with possibly different Fourier
matrix symbols in the Wiener-Hopf and in the Hankel operators. In the
second place, we consider these operators with equal Fourier symbols and
acting between weighted Lebesgue spaces Lp(R;w), where 1 < p < 1
and w belongs to a subclass of Muckenhoupt weights. In addition, singular
integral operators with Carleman shift and almost periodic coefficients are
also object of study. The main purpose of this thesis is to obtain regularity
properties characterizations of those classes of operators. By regularity
properties we mean those that depend on the kernel and cokernel of the
operator. The main techniques used are the equivalence relations between
operators and the factorization theory.
An invertibility characterization for the Wiener-Hopf-Hankel operators with
symbols belonging to the Wiener subclass of almost periodic functions APW
is obtained, assuming that a particular matrix function admits a numerical
range bounded away from zero and based on the values of a certain mean
motion. For Wiener-Hopf-Hankel operators acting between L2-spaces and with possibly
different AP symbols, criteria for the semi-Fredholm property and for
one-sided and both-sided invertibility are obtained and the inverses for all
possible cases are exhibited. For such results, a new type of AP factorization
is introduced.
Singular integral operators with Carleman shift and scalar almost periodic coefficients
are also studied. Considering an auxiliar and simpler operator, and
using appropriate factorizations, the dimensions of the kernels and cokernels
of those operators are obtained.
For Wiener-Hopf-Hankel operators with (possibly different) SAP and PAP
matrix symbols and acting between L2-spaces, criteria for the Fredholm
property are presented as well as the sum of the Fredholm indices of the
Wiener-Hopf plus Hankel and Wiener-Hopf minus Hankel operators.
By studying dependencies between different matrix Fourier symbols of
Wiener-Hopf plus Hankel operators acting between L2-spaces, results about
the kernel and cokernel of those operators are derived.
For Wiener-Hopf-Hankel operators acting between weighted Lebesgue
spaces, Lp(R;w), a study is made considering equal scalar Fourier symbols
in the Wiener-Hopf and in the Hankel operators and belonging to the
classes of APp;w, SAPp;w and PAPp;w. It is obtained an invertibility characterization
for Wiener-Hopf plus Hankel operators with APp;w symbols. In
the cases for which the Fourier symbols of the operators belong to SAPp;w
and PAPp;w, it is obtained semi-Fredholm criteria for Wiener-Hopf-Hankel
operators as well as formulas for the Fredholm indices of those operators.Nesta tese consideramos operadores de Wiener-Hopf-Hankel com símbolos
de Fourier nas classes das funções quase-periódicas, semi-quase periódicas
e quase periódicas por troços. Começamos por considerar operadores de
Wiener-Hopf-Hankel actuando entre espaços de Lebesgue L2 com símbolos
matriciais de Fourier possivelmente diferentes nos operadores de Wiener-
Hopf e de Hankel. Seguidamente, consideramos estes operadores com símbolos
de Fourier iguais actuando entre espaços de Lebesgue com pesos
Lp(R;w), onde 1 < p < 1 e w pertence a uma subclasse de pesos de
Muckenhoupt. Adicionalmente, são também objecto de estudo operadores
singulares integrais com deslocamento de Carleman e coeficientes quaseperiódicos.
O objectivo principal desta tese é obter caracterizações para tais
classes de operadores no que refere às suas propriedades de regularidade. Por
propriedades de regularidade nós designamos aquelas propriedades que dependem
do núcleo e do co-núcleo do operador. As principais técnicas usadas
são as relações de equivalência entre operadores e a teoria da factorização.
Uma caracterização da invertibilidade de operadores de Wiener-Hopf-Hankel
com símbolos pertencentes à subclasse de Wiener de funções quaseperiódicas
APW é obtida, assumindo que uma particular função matricial
admite um contradomínio numérico limitado fora de zero e baseando-nos
nos valores uma certa média de deslocamento. Para os operadores de Wiener-Hopf-Hankel actuando entre espaços de
Lebesgue L2 e com símbolos AP possivelmente diferentes, critérios para
a propriedade de semi-Fredholm e para a invertibilidade lateral e bi-lateral
são obtidos e inversos para todos os casos possíveis são apresentados. Com
vista a tais resultados, um novo tipo de factorização AP é introduzido.
Operadores singulares integrais com deslocamento de Carleman e com coeficientes
escalares quase-periódicos são também estudados. Considerando
um operador auxiliar mais simples e usando factorizações apropriadas, as
dimensões dos núcleos e dos co-núcleos destes operadores são obtidas.
Para operadores de Wiener-Hopf-Hankel com símbolos matriciais SAP e
PAP (possivelmente diferentes) actuando entre espaços de Lebesgue L2,
critérios para a propriedade de Fredholm são apresentados tal como a soma
dos índices de Fredholm dos operadores de Wiener-Hopf mais Hankel e
Wiener-Hopf menos Hankel.
Estudando dependências entre diferentes símbolos matriciais de Fourier dos
operadores de Wiener-Hopf mais Hankel actuando entre espaços de Lebesgue
L2, conclusões são obtidas acerca do núcleo e do co-núcleo destes operadores.
Para operadores de Wiener-Hopf-Hankel actuando entre espaços de Lebesgue
com pesos, Lp(R;w), é feito um estudo considerando símbolos de Fourier
escalares e iguais nos operadores de Wiener-Hopf e de Hankel e pertencentes
às classes APp;w, SAPp;w e PAPp;w. É obtida uma caracterização da invertibilidade
para operadores de Wiener-Hopf mais Hankel com símbolos APp;w.
No caso em que os símbolos de Fourier dos operadores pertencem a SAPp;w
e PAPp;w, são obtidos critérios de semi-Fredholm para os operadores de
Wiener-Hopf-Hankel assim como fórmulas para os índices de Fredholm de
tais operadores
Fredholm factorization of Wiener-Hopf scalar and matrix kernels
No full textA general theory to factorize the Wiener-Hopf (W-H) kernel using Fredholm Integral Equations (FIE) of the second kind is presented. This technique, hereafter called Fredholm factorization, factorizes the W-H kernel using simple numerical quadrature. W-H kernels can be either of scalar form or of matrix form with arbitrary dimensions. The kernel spectrum can be continuous (with branch points), discrete (with poles), or mixed (with branch points and poles). In order to validate the proposed method, rational matrix kernels in particular are studied since they admit exact closed form factorization. In the appendix a new analytical method to factorize rational matrix kernels is also described. The Fredholm factorization is discussed in detail, supplying several numerical tests. Physical aspects are also illustrated in the framework of scattering problems: in particular, diffraction problems. Mathematical proofs are reported in the pape
The Wiener-Hopf-Hilbert technique applied to problems in diffraction
No full textThis thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.A number of diffraction problems which have practical applications are examined using the Wiener-Hopf-Hilbert technique. Each problem is formulated as a matrix Wiener-Hopf equation, the solution of which requires the factor~sation of a matrix kernel. Since the determinant of the matrix kernel has poles in the cut plane, the
Wiener-Hopf-Hilbert technique is modified to allow the usual arguments to follow through. In each case an explicit matrix factorisation is carried out and asymptotic
expressions for the field scattered to infinity are obtained. The first problem solved is that of diffraction by a semi-infinite plane with different face impedances. The solution includes the case of an incident surface wave as well as an incident plane wave for an arbitrary angle of incidence. Graphs of the far-field are provided for various values of the half-plane impedance parameters. The second
problem examined is diffraction by a half-plane in a moving fluid. This is solved
without restriction on the impedance parameters of the half-plane and includes both the leading edge and trailing edge situations. The final problem is of radiation from an inductive wave-guide. Expressions are obtained for the field radiated at the waveguide mouth and the field reflected in the duct region.This work is funded by the UK Engineering and Physical Sciences Research Council (EPSRC
Subcritical Hopf bifurcations in a car-following model with reaction-time delay
No full textA nonlinear car-following model of highway traffic is considered, which includes the reaction-time delay of drivers. Linear stability analysis shows that the uniform flow equilibrium of the system loses its stability via Hopf bifurcations and thus oscillations can appear. The stability and amplitudes of the oscillations are determined with the help of normal-form calculations of the Hopf bifurcation that also handles the essential translational symmetry of the system. We show that the subcritical case of the Hopf bifurcation occurs robustly, which indicates the possibility of bistability. We also show how these oscillations lead to spatial wave formation as can be observed in real-world traffic flow
When Hopf Algebras are Frobenius Algebras
No full textAbstractR. Larson and M. Sweedler recently proved that for free finitely generated Hopf algebras H over a principal ideal domain R the following are equivalent: (a) H has an antipode and (b) H has a nonsingular left integral. In this paper I give a generalization of this result which needs only a minor restriction, which, for example, always holds if pic(R) = 0 for the base ring R. A finitely generated projective Hopf algebra H over R has an antipode if and only if H is a Frobenius algebra with a Frobenius homomorphism ψ such that Σ h(1) ψ(h(2)) = ψ(h) · 1 for all h ϵ H. We also show that the antipode is bijective and that the ideal of left integrals is a free rank 1, R-direct summand of H
Global Hopf bifurcation in the ZIP regulatory system
No full textRegulation of zinc uptake in roots of Arabidopsis thaliana has recently been modeled by a system of ordinary differential equations based on the uptake of zinc, expression of a transporter protein and the interaction between an activator and inhibitor. For certain parameter choices the steady state of this model becomes unstable upon variation in the external zinc concentration. Numerical results show periodic orbits emerging between two critical values of the external zinc concentration. Here we show the existence of a global Hopf bifurcation with a continuous family of stable periodic orbits between two Hopf bifurcation points. The stability of the orbits in a neighborhood of the bifurcation points is analyzed by deriving the normal form, while the stability of the orbits in the global continuation is shown by calculation of the Floquet multipliers. From a biological point of view, stable periodic orbits lead to potentially toxic zinc peaks in plant cells. Buffering is believed to be an efficient way to deal with strong transient variations in zinc supply. We extend the model by a buffer reaction and analyze the stability of the steady state in dependence of the properties of this reaction. We find that a large enough equilibrium constant of the buffering reaction stabilizes the steady state and prevents the development of oscillations. Hence, our results suggest that buffering has a key role in the dynamics of zinc homeostasis in plant cells
Hopf bifurcation in a gene regulatory network model : molecular movement causes oscillations
No full textM.A.J.C. and M.S. gratefully acknowledge the support of the ERC Advanced Investigator Grant 227619, “M5CGS — From Mutations to Metastases: Multiscale Mathematical Modelling of Cancer Growth and Spread”. M.S. would also like to thank the support from the Mathematical Biosciences Institute at the Ohio State University and NSF Grant DMS0931642.Gene regulatory networks, i.e. DNA segments in a cell which interact with each other indirectly through their RNA and protein products, lie at the heart of many important intracellular signal transduction processes. In this paper, we analyze a mathematical model of a canonical gene regulatory network consisting of a single negative feedback loop between a protein and its mRNA (e.g. the Hes1 transcription factor system). The model consists of two partial differential equations describing the spatio-temporal interactions between the protein and its mRNA in a one-dimensional domain. Such intracellular negative feedback systems are known to exhibit oscillatory behavior and this is the case for our model, shown initially via computational simulations. In order to investigate this behavior more deeply, we undertake a linearized stability analysis of the steady states of the model. Our results show that the diffusion coefficient of the protein/mRNA acts as a bifurcation parameter and gives rise to a Hopf bifurcation. This shows that the spatial movement of the mRNA and protein molecules alone is sufficient to cause the oscillations. Our result has implications for transcription factors such as p53, NF-κB and heat shock proteins which are involved in regulating important cellular processes such as inflammation, meiosis, apoptosis and the heat shock response, and are linked to diseases such as arthritis and cancer.Peer reviewe
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