Spelling suggestions: "subject:"integrals"" "subject:"ntegrals""
91 
Period integrals and other direct images of DmodulesTveiten, Ketil January 2015 (has links)
This thesis consists of three papers, each touching on a different aspect of the theory of rings of differential operators and Dmodules. In particular, an aim is to provide and make explicit good examples of Dmodule directimages, which are all but absent in the existing literature.The first paper makes explicit the fact that Bsplines (a particular class of piecewise polynomial functions) are solutions to Dmodule theoretic direct images of a class of Dmodules constructed from polytopes.These modules, and their direct images, inherit all the relevant combinatorial structure from the defining polytopes, and as such are extremely wellbehaved.The second paper studies the ring of differential operator on a reduced monomial ring (aka. StanleyReisner ring), in arbitrary characteristic.The twosided ideal structure of the ring of differential operators is described in terms of the associated abstract simplicial complex, and several quite different proofs are given.The third paper computes the monodromy of the period integrals of Laurent polynomials about the singular point at the origin. The monodromy is describable in terms of the Newton polytope of the Laurent polynomial, in particular the combinatorialalgebraic operation of mutation plays an important role. Special attention is given to the class of maximally mutable Laurent polynomials, as these are one side of the conjectured correspondance that classifies Fano manifolds via mirror symmetry. / <p>At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: Accepted. Paper 2: Manuscript. Paper 3: Manuscript.</p>

92 
Symmetry properties for first integralsMahomed, Komal Shahzadi 02 February 2015 (has links)
A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. July 2014. / This is the study of Lie algebraic properties of first integrals of scalar second, third and
higherorder ordinary differential equations (ODEs). The Lie algebraic classification of such differential equations is now wellknown from the works of Lie [10] as
well as recently Mahomed and Leach [19]. However, the algebraic properties of first
integrals are not known except in the maximal cases for the basic first integrals and
some of their quotients. Here our intention is to investigate the complete problem for
scalar secondorder and maximal symmetry classes of higherorder ODEs using Lie
algebras and Lie symmetry methods. We invoke the realizations of lowdimensional
Lie algebras.
Symmetries of the fundamental first integrals for scalar secondorder ODEs which are
linear or linearizable by point transformations have already been obtained. Firstly we
show how one can determine the relationship between the point symmetries and the
first integrals of linear or linearizable scalar ODEs of order two. Secondly, a complete
classi cation of point symmetries of first integrals of such linear ODEs is studied. As a
consequence, we provide a counting theorem for the point symmetries of first integrals
of scalar linearizable secondorder ODEs. We show that there exists the 0, 1, 2 or 3
point symmetry cases. It is proved that the maximal algebra case is unique.
By use of Lie symmetry group methods we further analyze the relationship between the
first integrals of the simplest linear thirdorder ODEs and their point symmetries. It
is wellknown that there are three classes of linear thirdorder ODEs for maximal and
submaximal cases of point symmetries which are 4, 5 and 7. The simplest scalar linear
thirdorder equation has seven point symmetries. We obtain the classifying relation
between the symmetry and the first integral for the simplest equation. It is shown
that the maximal Lie algebra of a first integral for the simplest equation y000 = 0 is
unique and fourdimensional. Moreover, we show that the Lie algebra of the simplest
linear thirdorder equation is generated by the symmetries of the two basic integrals.
We also obtain counting theorems of the symmetry properties of the first integrals for
such linear thirdorder ODEs of maximal type. Furthermore, we provide insights into
the manner in which one can generate the full Lie algebra of higherorder ODEs of
maximal symmetry from two of their basic integrals.
The relationship between rst integrals of submaximal linearizable thirdorder ODEs
and their symmetries are investigated as well. All scalar linearizable thirdorder equations
can be reduced to three classes by point transformations. We obtain the
classifying relations between the symmetries and the first integral for submaximal
cases of linear thirdorder ODEs. It is known, from the above, that the maximum Lie
algebra of the first integral is achieved for the simplest equation. We show that for
the other two classes they are not unique. We also obtain counting theorems of the
symmetry properties of the rst integrals for these classes of linear thirdorder ODEs.
For the 5 symmetry class of linear thirdorder ODEs, the first integrals can have 0,
1, 2 and 3 symmetries and for the 4 symmetry class of linear thirdorder ODEs they
are 0, 1 and 2 symmetries respectively. In the case of submaximal linear higherorder
ODEs, we show that their full Lie algebras can be generated by the subalgebras of
certain basic integrals. For the n+2 symmetry class, the symmetries of the rst integral
I2 and a twodimensional subalgebra of I1 generate the symmetry algebra and for
the n + 1 symmetry class, the full algebra is generated by the symmetries of I1 and a
twodimensional subalgebra of the quotient I3=I2.
Finally, we completely classify the first integrals of scalar nonlinear secondorder ODEs
in terms of their Lie point symmetries. This is performed by first obtaining the classifying
relations between point symmetries and first integrals of scalar nonlinear second order
equations which admit 1, 2 and 3 point symmetries. We show that the maximum
number of symmetries admitted by any first integral of a scalar secondorder nonlinear
(which is not linearizable by point transformation) ODE is one which in turn provides
reduction to quadratures of the underlying dynamical equation. We provide physical
examples of the generalized EmdenFowler, LaneEmden and modi ed Emden equations.

93 
A variational effective potential approximation for the Feynman path integral approach to statistical mechanics.January 1992 (has links)
by Lee Siukeung. / Parallel title in Chinese. / Thesis (M.Phil.)Chinese University of Hong Kong, 1992. / Includes bibliographical references (leaves 162164). / Chapter Chapter 1  Introduction  p.5 / Chapter Chapter 2  Path Integrals / Chapter 2.1  Path´ؤIntegral Approach to Quantum Mechanics  p.8 / Chapter 2.2  Path´ؤIntegral Approach to Statistical Mechanics  p.14 / Chapter 2.3  Variational Principle  p.18 / Chapter 2.4  "Variational Method Proposed by Giachetti and Tognetti, and by Feynman and Kleinert" / Chapter 2.4.1  Effective Classical Partition Function  p.24 / Chapter 2.4.2  Particle Distribution Function From Effective Classical Potential  p.34 / Chapter Chapter 3  Systematic Perturbation Corrections to the Variational Approximation Proposed in Section2.4 / Chapter 3.1  Formalism / Chapter 3.1.1  Free Energy  p.38 / Chapter 3.1.2  Particle Distribution Function  p.49 / Chapter 3.2  Second Order Correction to Free Energy  p.53 / Chapter 3.3  First Order Correction to Particle Distribution Function  p.60 / Chapter Chapter 4  Examples and Results / Chapter 4.1  Quartic Anharmonic Oscillator / Chapter 4.1.1  "Free Energy, Internal Energy and Specific Heat"  p.69 / Chapter 4.1.2  Particle Distribution Function  p.87 / Chapter 4.2  Symmetric Doublewell Potential / Chapter 4.2.1  "Free Energy, Internal Energy and Specific Heat"  p.88 / Chapter 4.2.2  Particle Distribution Function  p.106 / Chapter 4.3  Quarticcubic Anharmonic Potential / Chapter 4.3.1  Free Energy  p.108 / Chapter 4.3.2  Particle Distribution Function  p.115 / Chapter Chapter 5  Application to the Onedimensional GinzburgLandau Model / Chapter 5.1  Introduction  p.120 / Chapter 5.2  Exact Partition Function and Free Energy Per Unit Length  p.123 / Chapter 5.3  Zeroth Order Approximation to Free Energy Per Unit Length  p.126 / Chapter 5.4  Exact Specific Heat  p.133 / Chapter 5.5  Zeroth Order Approximation to Specific Heat  p.139 / Chapter Chapter 6  Conclusion  p.141 / Chapter Appendix I  Functional Calculus  Differentiation  p.145 / Chapter Appendix II  Evaluation of Feynman Propagator Δf(τ)  p.147 / Chapter Appendix III  Vanishing of the First Order Correctionβf1  p.150 / Chapter Appendix IV  Numerical Method for the Energy Eigenvalues and Eigenfunctions of the Onedimensional Schroedinger Equation with ax2 + bx4 Potential  p.153 / Chapter Appendix V  Numerical Integrations with imaginary Ω  p.158 / References  p.162 / Figures  p.165

94 
Wavelets and singular integral operators.January 1999 (has links)
by Lau Shuikong, Francis. / Thesis (M.Phil.)Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 9598). / Abstracts in English and Chinese. / Chapter 1  General Theory of Wavelets  p.8 / Chapter 1.1  Introduction  p.8 / Chapter 1.2  Multiresolution Analysis and Wavelets  p.9 / Chapter 1.3  Orthonormal Bases of Compactly Supported Wavelets  p.12 / Chapter 1.3.1  Example : The Daubechies Wavelets  p.15 / Chapter 1.4  Wavelets in Higher Dimensions  p.20 / Chapter 1.4.1  Tensor product method  p.20 / Chapter 1.4.2  Multiresolution Analysis in Rd  p.21 / Chapter 1.5  Generalization to frames  p.25 / Chapter 2  Wavelet Bases Numerical Algorithm  p.27 / Chapter 2.1  The Algorithm in Wavelet Bases  p.27 / Chapter 2.1.1  Definitions and Notations  p.28 / Chapter 2.1.2  Fast Wavelet Transform  p.31 / Chapter 2.2  WaveletBased Quadratures  p.33 / Chapter 2.3  "The Integral Operator, Standard and Nonstandard Form"  p.39 / Chapter 2.3.1  The Standard Form  p.40 / Chapter 2.3.2  The Nonstandard Form  p.41 / Chapter 2.4  The CalderonZygmund Operator and Numerical Cal culation  p.45 / Chapter 2.4.1  Numerical Algorithm to Construct the Non standard Form  p.45 / Chapter 2.4.2  Numerical Calculation and Compression of Op erators  p.45 / Chapter 2.5  Differential Operators in Wavelet Bases  p.48 / Chapter 3  T(l)Theorem of David and Journe  p.55 / Chapter 3.1  Definitions and Notations  p.55 / Chapter 3.1.1  T(l) Operator  p.56 / Chapter 3.2  The Wavelet Proof of the T(l)Theorem  p.59 / Chapter 3.3  Proof of the T(l)Theorem (Continue)  p.64 / Chapter 3.4  Some recent results on the T(l)Theorem  p.70 / Chapter 4  Singular Values of Compact Pseudodifferential Op erators  p.72 / Chapter 4.1  Background  p.73 / Chapter 4.1.1  Singular Values  p.73 / Chapter 4.1.2  Schatten Class Ip  p.73 / Chapter 4.1.3  The Ambiguity Function and the Wigner Dis tribution  p.74 / Chapter 4.1.4  Weyl Correspondence  p.76 / Chapter 4.1.5  Gabor Frames  p.78 / Chapter 4.2  Singular Values of Lσ  p.82 / Chapter 4.3  The CalderonVaillancourt Theorem  p.87 / Chapter 4.3.1  HolderZygmund Spaces  p.87 / Chapter 4.3.2  Smooth Dyadic Resolution of Unity  p.88 / Chapter 4.3.3  The proof of the CalderonVaillancourt The orem  p.89 / Bibliography

95 
Matsubara dynamics and its practical implementationWillatt, Michael John January 2017 (has links)
This thesis develops a theory for approximate quantum timecorrelation functions, Matsubara dynamics, that rigorously describes how to combine quantum statistics with classical dynamics. Matsubara dynamics is based on Feynman's path integral formulation of quantum mechanics and is expected to describe the physics of any system that satisfies quantum Boltzmann statistics and exhibits rapid quantum decoherence, e.g. liquid water at room temperature. Having derived the Matsubara dynamics theory and explored the symmetry properties that it shares with the quantum Kubo timecorrelation function, we demonstrate that two heuristic computational methods, Centroid Molecular Dynamics and Ring Polymer Molecular Dynamics, are based on quantifiable approximations to the Matsubara dynamics timecorrelation function. This provides these methods with a stronger theoretical foundation and helps to explain their strengths and shortcomings. We then apply the Matsubara dynamics theory to a recently developed computational method of Poulsen et al. called the planetary model. We show that the planetary model is based on a harmonic approximation to Matsubara dynamics that is engineered to maintain the conservation of the quantum Boltzmann distribution, so quantum statistics and classical dynamics remain harmonised. By making practical modifications to the planetary model, we were able to calculate infrared absorption spectra for a point charge model of condensedphase water over a range of thermodynamic conditions. We find that this harmonic approximation to Matsubara dynamics provides a good description of bending and vibrational motions and is expected to be a useful tool for future spectroscopic studies of more complex, polarisable models of water.

96 
Singular integral operators on amalgam spaces. / CUHK electronic theses & dissertations collectionJanuary 2004 (has links)
by HonMing Ho. / "May 2004." / Thesis (Ph.D.)Chinese University of Hong Kong, 2004. / Includes bibliographical references (p. 6971). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

97 
A Survey in Mean Value TheoremsNeuser, David A. 01 May 1970 (has links)
A variety of new mean value theorems are presented along with interesting proofs and generalizations of the standard theorems.
Three proofs are given for the ordinary Mean Value Theorem for derivatives, the third of which is interesting in that it is independent of of Rolle's Theorem. The Second Mean Value Theorem for derivatives is generalized, with the use of determinants, to three functions and also generalized in terms of nth order derivatives.
Observing that under certain conditions the tangent line to the curve of a differentiable function passes through the initial point, we find a new type of mean value theorem for derivatives. This theorem is extended to two functions and later in the paper an integral analog is given together with integral mean value theorems.
Many new mean value theorems are presented in their respective settings including theorems for the total variation of a function, the arc length of the graph of a function, and for vectorvalued functions. A mean value theorem in the complex plane is given in which the difference quotient is equal to a linear combination of the values of the derivative. Using a regular derivative, the ordinary Mean Value Theorem for derivatives is extended into Rn, n>1.

98 
New developments in the construction of lattice rules: applications of lattice rules to highdimensional integration problems from mathematical finance.Waterhouse, Benjamin James, School of Mathematics, UNSW January 2007 (has links)
There are many problems in mathematical finance which require the evaluation of a multivariate integral. Since these problems typically involve the discretisation of a continuous random variable, the dimension of the integrand can be in the thousands, tens of thousands or even more. For such problems the Monte Carlo method has been a powerful and popular technique. This is largely related to the fact that the performance of the method is independent of the number of dimensions. Traditional quasiMonte Carlo techniques are typically not independent of the dimension and as such have not been suitable for highdimensional problems. However, recent work has developed new types of quasiMonte Carlo point sets which can be used in practically limitless dimension. Among these types of point sets are Sobol' sequences, Faure sequences, NiederreiterXing sequences, digital nets and lattice rules. In this thesis, we will concentrate on results concerning lattice rules. The typical setting for analysis of these new quasiMonte Carlo point sets is the worstcase error in a weighted function space. There has been much work on constructing point sets with small worstcase errors in the weighted Korobov and Sobolev spaces. However, many of the integrands which arise in the area of mathematical finance do not lie in either of these spaces. One common problem is that the integrands are unbounded on the boundaries of the unit cube. In this thesis we construct function spaces which admit such integrands and present algorithms to construct lattice rules where the worstcase error in this new function space is small. Lattice rules differ from other quasiMonte Carlo techniques in that the points can not be used sequentially. That is, the entire lattice is needed to keep the worstcase error small. It has been shown that there exist generating vectors for lattice rules which are good for many different numbers of points. This is a desirable property for a practitioner, as it allows them to keep increasing the number of points until some error criterion is met. In this thesis, we will develop fast algorithms to construct such generating vectors. Finally, we apply a similar technique to show how a particular type of generating vector known as the Korobov form can be made extensible in dimension.

99 
Simple Layer Potentials on Lipschitz Surfaces: An Asymptotic ApproachThim, Johan January 2009 (has links)
This work is devoted to the equation <img src="http://www.divaportal.org/cgibin/mimetex.cgi?%5Cint_%7BS%7D%0A%5Cfrac%7Bu(y)%20%5C,%20dS(y)%7D%7B%7Cxy%7C%5E%7BN1%7D%7D%20=%20f(x)%20%5Ctext%7B,%7D%20%5Cqquad%20%5Cqquad%20x%20%5Cin%20S%20%5Ctext%7B,%7D%0A%5Cqquad%20%5Cqquad%20%5Cqquad%20%5Cqquad%20%5Cqquad%20%5Cqquad%20%5Cqquad%20(1)%0A" /> where S is the graph of a Lipschitz function φ on RN with small Lipschitz constant, and dS is the Euclidian surface measure. The integral in the lefthand side is referred to as a simple layer potential and f is a given function. The main objective is to find a solution u to this equation along with estimates for solutions near points on S. Our analysis is carried out in local Lpspaces and local Sobolev spaces, and the estimates are given in terms of seminorms. In Paper 1, we consider the case when S is a hyperplane. This gives rise to the classical Riesz potential operator of order one, and we prove uniqueness of solutions in the largest class of functions for which the potential in (1) is defined as an absolutely convergent integral. We also prove an existence result and derive an asymptotic formula for solutions near a point on the surface. Our analysis allows us to obtain optimal results concerning the class of righthand sides for which a solution to (1) exists. We also apply our results to weighted Lp and Sobolev spaces, showing that for certain weights, the operator in question is an isomorphism between these spaces. In Paper 2, we present a fixed point theorem for a locally convex space <img src="http://www.divaportal.org/cgibin/mimetex.cgi?%5Cmathscr%7BX%7D" />, where the topology is given by a family <img src="http://www.divaportal.org/cgibin/mimetex.cgi?%5C%7Bp(%20%5C,%20%5Ccdot%20%5C,%20;%20%5Calpha%20)%5C%7D_%7B%5Calpha%20%5Cin%20%5COmega%7D" /> of seminorms. We study the existence and uniqueness of fixed points for a mapping<img src="http://www.divaportal.org/cgibin/mimetex.cgi?%5Cmathscr%7BK%7D%20%5C,%20:%20%5C;%20%5Cmathscr%7BD_K%7D%20%5Crightarrow%20%5Cmathscr%7BD_K%7D" /> defined on a set <img src="http://www.divaportal.org/cgibin/mimetex.cgi?%5Cmathscr%7BD_K%7D%20%5Csubset%20%5Cmathscr%7BX%7D" />. It is assumed that there exists a linear and positive operator K, acting on functions defined on the index set Ω, such that for every <img src="http://www.divaportal.org/cgibin/mimetex.cgi?u,v%20%5Cin%20%5Cmathscr%7BD_K%7D" />, <img src="http://www.divaportal.org/cgibin/mimetex.cgi?p(%5Cmathscr%7BK%7D(u)%20%20%5Cmathscr%7BK%7D(v)%20%5C,%20;%20%5C,%20%5Calpha%20)%20%0A%5Cleq%20K(p(uv%20%5C,%20;%20%5C,%20%5Ccdot%20%5C,%20))%20(%5Calpha)%20%5Ctext%7B,%7D%20%5Cqquad%20%5Cqquad%20%5Calpha%20%5Cin%20%5COmega%0A%5Ctext%7B.%7D%0A" /> Under some additional assumptions, one of which is the existence of a fixed point for the operator K + p(<img src="http://www.divaportal.org/cgibin/mimetex.cgi?%5Cmathscr%7BK%7D(0)" /> ; · ), we prove that there exists a fixed point of <img src="http://www.divaportal.org/cgibin/mimetex.cgi?%5Cmathscr%7BK%7D" />. For a class of elements satisfying Kn (p(u ; · ))(α) → 0 as n → ∞, we show that fixed points are unique. This class includes, in particular, the solution we construct in the paper. We give several applications, proving existence and uniqueness of solutions for two types of first and second order nonlinear differential equations in Banach spaces. We also consider pseudodifferential equations with nonlinear terms. In Paper 3, we treat equation (1) in the case when S is a general Lipschitz surface and 1 < p < ∞. Our results are presented in terms of Λ(r), which is the Lipschitz constant of φ on the ball centered at the origin with radius 2r. Estimates of solutions to (1) are provided, which can be used to obtain knowledge about behaviour near a point on S in terms of seminorms. We also show that solutions to (1) are unique if they are subject to certain growth conditions. Examples are given when specific assumptions are placed on Λ. The main tool used for both existence and uniqueness is the fixed point theorem from Paper 2. In Paper 4, we collect some properties and estimates of Riesz potential operators, and also for the operator that was used in Paper 1 and Paper 3 to invert the Riesz potential of order one on RN, for the case when the density function is either radial or has mean value zero on spheres. It turns out that these properties define invariant subspaces of the respective domains of the operators in question.

100 
A Third Order Numerical Method for Doubly Periodic Electromegnetic ScatteringNicholas, Michael J 31 July 2007 (has links)
We here developed a thirdorder accurate numerical method for scattering of 3D electromagnetic waves by doubly periodic structures. The method is an intuitively simple numerical scheme based on a boundary integral formulation. It involves smoothing the singular Green's functions in the integrands and finding correction terms to
the resulting smooth integrals. The analytical method is based on the singular integral methods of J. Thomas Beale, while the scattering problem is motivated by the 2D work of Stephanos Venakides, Mansoor Haider, and Stephen Shipman. The 3D problem was done with boundary element methods by Andrew Barnes. We present a method that is both more straightforward and more accurate. In solving these problems, we have used the M\"{u}ller integral equation formulation of Maxwell's equations, since it is a Fredholm integral equation of the second kind and is wellposed. M\"{u}ller derived his equations for the case of a compact scatterer. We outline the derivation and adapt it to a periodic scatterer. The periodic Green's functions found in the integral equation contain singularities which make it difficult to evaluate them numerically with accuracy. These functions are also very time consuming to evaluate numerically. We use Ewald splitting to represent these functions in a way that can be computed rapidly.We present a method of smoothing the singularity of the Green's function while maintaining its periodicity. We do local analysis of the singularity in order to identify and eliminate the largest sources of error introduced by this smoothing. We prove that with our derived correction terms, we can replace the singular integrals with smooth integrals and only introduce a error that is third order in the grid spacing size. The derivation of the correction terms involves transforming to principal directions using concepts from differential geometry. The correction terms are necessarily invariant under this transformation and depend on geometric properties of the scatterer such as the mean curvature and the differential of the Gauss map. Able to evaluate the integrals to a higher order, we implement a \mbox{GMRES} algorithm to approximate solutions of the integral equation. From these solutions, M\"{u}ller's equations allow us to compute the scattered fields and transmission coefficients. We have also developed acceleration techniques that allow for more efficient computation.We provide results for various scatterers, including a test case for which exact solutions are known. The implemented method does indeed converge with third order accuracy. We present results for which the method successfully resolves Wood's anomaly resonances in transmission. / Dissertation

Page generated in 0.043 seconds