# Integral equations

## Top PDF Integral equations: ### Banach space valued stochastic integral equations and their optimal control

In chapter 4 the duality principles between forward and backward stochastic Volterra integral equations are discussed. The forward stochastic Volterra integral equation (4.1) is defined in a σ-finite space where q ≥ 2 and correspondingly by using its adjoint operators, two types of backward Volterra integral equations in σ-finite space with q ≤ 2 are introduced. First the duality principle between Forward and bachward Volltera Equations (4.1), (4.2) are proved in Theorem 4.1. The second and more general duality principle are also given in Theorem 4.2 that is between Equations (4.1) and (4.3). The Itô formula in Banach spaces in  are very useful to compute duality principles and as in Remark 4.1 is explained these computation for duality principle hold for every UMD-spaces if forward stochastic Volterra integral equations and corresponding backward stochastic Volterra integral equations are well-defined and more importantly martingale representation theorem can be used efficiently, for example in co-type(2) spaces.
Mehr anzeigen ### Hybrid Type Method of Numerical Solution Integral Equations and its Applications

How to reference In order to correctly reference this scholarly work, feel free to copy and paste the following: D.G. Arsenjev, V.M. Ivanov and N.A. Berkovskiy (2011). Hybrid Type Method of Numerical Solution Integral Equations and its Applications, Numerical Simulations - Applications, Examples and Theory, Prof. Lutz Angermann (Ed.), ISBN: 978-953-307-440-5, InTech, Available from: ### Time Domain Boundary Integral Equations Analysis

The energy dissipation due to numerical integration methods can lead to misleading results for large systems that need to be iterated for long time intervals. The solution is to use symplectic space-time integration methods that comply the Lorentz reciprocity theorem and fully conserve the energy. In this regard, the advantage of symmetric dis- cretization in numerical solution of retarded functional integral equations is not so far clear. In the TDIE, the past solution samples contribute in determination of present sta- tus of the system, and hence, the solution accuracy is coupled to the late-time stability. Stability analysis of numerical methods is typically carried out based on the error bound of the approximate solution without considering the energy balance. To avoid the error prop- agation of asymmetric quadrature routines leading to unbounded energy, in this section the numerical calculation of the double surface integrals is studied in a totally symmetric way where the number of quadrature points are adaptively refined by simultaneous partitioning of source and observation subdomains. Probing the eigenvalue spectra of the retarded time dependent system reveals that many unwanted modes cause the propagation of numerical errors that may not even unveil the late-time instabilities due to considerations in spatial discretization. Among choices of the stable time integration methods, the one through which the EM energy is conserved are sought in the next section.
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200 Mehr lesen ### Electromagnetic Wave Scattering at Biperiodic Surfaces: Variational Formulation, Boundary Integral Equations and High Order Solvers

In Chapter 5, the numerical scheme is presented. As mentioned above, it is a variant of the method from [19, 20] and constitutes an improvement of the scheme in , which consists in a reduction of the overall complexity by introducing another orthogonal projection. A key tool is the removal of the weak singularity by a transformation into polar coordinates for the first integral in (1.2). As a consequence, the corresponding integral operator takes on a non-standard form making the analysis of its mapping properties, as well as for its approximation, technically complicated. Therefore, the approach is demonstrated at first on single integral equations and later generalized to systems. The scheme is a collocation method and achieves super-algebraic convergence rate (provided the surface is smooth). As another novelty, the analysis yields an explicit dependence of the constants in the stability and convergence estimates on some number % which couples the support of various cut-off functions to each other. The results were developed in collaboration with Tilo Arens and prepublished in . Therein, the application to typical boundary value problems such as potential and scattering problems both for bounded obstacles and for biperiodic surfaces is emphasized and numerical examples are presented which demonstrate the expected convergence rates in practice.
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347 Mehr lesen ### Nonlinear Integral Equations for the Thermodynamics of Integrable Quantum Chains

In this thesis, we have treated the thermodynamics of various integrable quantum chains of Uimin-Sutherland type on the basis of nonlinear integral equations (NLIEs). There exist several approaches to derive such NLIEs, confer [34, 90]. Nevertheless, only the approach developed in [40, 41] is known to allow for an efficient numerical solution for arbitrary finite temperature and rather general chemical potentials. The approach is based on a Trotter- Suzuki mapping of the quantum chain to an equivalent two-dimensional classical model. In case of the Uimin-Sutherland model the classical counterpart is an inhomogeneous Perk- Schultz model, where alternating R matrices appear in vertical direction. For the latter model one defines the so-called quantum transfer matrix (QTM), which is basically the inhomogeneous column-to-column transfer matrix with an additional spectral parameter. This allows for the formulation of the partition function of the Uimin-Sutherland model solely in terms of the QTM. The big advantage is that the thermodynamical limit can be performed exactly, and it follows that the thermodynamical properties of the model depend on just the largest eigenvalue of the QTM. Moreover, the QTM can be diagonalized by means of the Bethe ansatz, since the underlying R matrices are solutions to the Yang-Baxter equation and thus a whole commuting family of QTMs exists. The calculation of the Bethe ansatz roots still turns out to be problematic. Therefore, the Bethe ansatz equations are encoded into a set of suitable auxiliary functions, which turn out to be determined by a finite set of coupled NLIEs containing only convolution-type integrals. For numerical investigations, the NLIEs can then be solved efficiently by iteration, where the fast Fourier transform is used to calculate the convolutions. In contrast to the other approaches, however, no general construction of the NLIEs for arbitrary cases of the Uimin-Sutherland model is known. Previous to our work, results were known for up to three components at most [27, 36, 37].
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149 Mehr lesen ### Discretization of backward stochastic Volterra integral equations

In this chapter we briefly introduce the main concepts from stochastic analysis which are the basement for our research problem in this thesis. In Section 1.1 we present some well-known definitions and basic results from the theory of probabil- ity. Section 1.2 is devoted to the well-posedness problems for backward stochastic differential equations (BSDEs, for short) and backward stochastic Volterra inte- gral equations (BSVIE, for short), which are the main research object of this work. At last, in Section 1.3 we formulate the main result of this thesis. ### Curvature Detection by Integral Transforms

for almost all x ∈ R 2 is called conformal monogenic signal. Remark 2.21. It is unclear, how the Riesz transform H i [g m x,s ] is exactly defined in the original paper [FWS11]. Since g m x,s is supported on a two-dimensional surface due to Remark 2.19, which is a Lebesgue null set, the Riesz transform of g m x,s ought to be 0. A similar case appears in [FWS11, p. 314] for the Radon transform. There the problem of vanishing integral transform was swiftly circumnavigated by considering a Radon measure on the lower dimensional manifold S 2 . “Let P denote a plane in R 3 with C = P ∩ S 2 such that C 6= ;. If we integrate over the plane P, we actually want to integrate g m x,s over the circle C. Since this is a Lebesgue null set with respect to the standard Lebesgue measure in the plane we have to introduce an alternate measure. In- stead, the Radon transform has to be understood with respect to the Radon measure δ(C(u))du where C(u) = 0 ⇔ u ∈ C . . . ”
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202 Mehr lesen ### Synthese von Blockcopolymeren zur Darstellung integral asymmetrischer Membranen

Polystyrol-block-poly(2-vinylpyridin) ist ein nicht-ionisches amphiphiles Blockcopolymer. Es sollte sich daher sehr gut für die Darstellung von Membranen mit Hilfe des Phaseninver- sionsprozesses eignen. Mit P2VP besitzt es zudem einen pH-sensitiven Block. Zu Beginn des Projekts, in dessen Rahmen die vorliegende Arbeit angefertigt wurde, wurde jedoch an- genommen, dass die Darstellung einer integral asymmetrischen Membran aus PS-b-P2VP mittels Phaseninversion nicht möglich sei. Aufgrund des geringeren χ-Parameters von PS- b-P2VP im Vergleich zu PS-b-P4VP sollte daher die Synthese von PS-b-P(2VP-ran-4VP) untersucht werden. Dies sollte die Einbindung von P2VP in ein Membran bildendes Poly- mer ermöglichen. Ließen sich die Poren einer Membran bestehend aus P2VP und P4VP bei unterschiedlichen pH-Werten schließen, wäre zusätzlich eine zweistufig schaltbare Membran entstanden. Beide Annahmen konnten mit Ergebnissen die im Rahmen des Projekts hervor- gingen nicht bestätigt werden, sodass von weiteren Experimenten sowohl synthetischer, als auch membrantechnischer Natur abgesehen wurde. Zusätzlich wurde zur Überprüfung der zuvor erwähnten Annahme die Membranherstellung von PS-b-P2VP mittels Phaseninversion untersucht. Die daraus hervorgegangenen Ergebnisse sind in Abschn. 4.2.3 beschrieben. Auch PS-b-P4VP ist ebenso wie PS-b-P2VP ein amphiphiles Blockcopolymer. PS-b-P4VP wurde hinsichtlich der Darstellung von integral asymmetrischen Membranen mit Hilfe des Phaseninversionsprozesses bereits vielfach untersucht [5,6] und wird in dieser Arbeit daher nicht weiter behandelt. Die in dieser Arbeit dargestellten PS-b-P4VP-Polymere wurden im Rahmen des EU-Projekts „SELFMEM - Self-assembled Polymer Membranes“ von Team- kollegen bei der Membranherstellung untersucht. Hierbei wurde z.B. der Einfluss von Addi- tiven auf die Strukturbildung von Membranen ermittelt. [12,85] Die Untersuchungen ergaben eine Verbesserung der Membranoberflächenstruktur aufgrund der Zugabe von Magnesium- acetat bzw. Cyclodextrin.
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167 Mehr lesen There are some difficulties in interpreting the beha­ vioural assum ptions underlying the determination of ex­ ports. As we have noted above, it is not clear whether the volum e equations are to be interpreted as demand equa­ tions or reduced from relationships. If they can be treated as dem and equations then the export price relationships m ight be reasonably regarded as supply equations. However if the volum e equation is a reduced form then presum ably the price equation should be derived from the sam e behavioural considerations. This raises a potential problem of internal consistency. If, for example, the underlying hypothesis were of a sm all economy trading in a hom ogeneous good then we would expect the price elasticity of dem and to be infinite (as measured by the price coefficient in the volum e equation) and this should coincide
Mehr anzeigen ### Randschichtmodifikation von integral verzweigten Blechprofilen mit UFG Gradientengefügen

Spaltprofilieren sowie Spaltbiegen sind innovative Verfahren zur kontinuierlichen Herstellung integral verzweigter Blechstrukturen. Die aus Blechen hergestellten, ver- zweigten Strukturen sind insbesondere für Leichtbauanwendungen interessant, da durch die Verzweigungen eine hohe Steifigkeit bei gleichzeitig geringem Bauteilge- wicht erreicht werden kann. Ein weiterer Vorteil der Verfahren Spaltprofilieren und Spaltbiegen ist die prozessimmanente Entstehung ultrafeinkörniger (UFG) Gefüge. Die durch das Spaltprofilieren bzw. Spaltbiegen erzeugten UFG Gefüge weisen eine ge- längte, pancake-förmige Kornform auf und liegen an den gespaltenen Bauteiloberflä- chen bis in eine Tiefe von 300 µm bzw. 500 µm vor. Mit den ultrafeinkörnigen Gefü- gen geht eine Verdopplung der Härte und Festigkeit sowie eine deutliche Erhöhung der Dauerfestigkeit einher, verglichen zu dem Blechmaterial im unverformten Aus- gangszustand. Durch die sehr guten mechanischen Eigenschaften der UFG Gefüge und durch die Form der Spaltprofile eignen sich spaltprofilierte Bauteile insbesondere für den Einsatz als Linearführung. Spaltgebogene Profile können in Stringerprofile umge- formt werden und bieten so die Möglichkeit, beispielsweise im Bereich der Luftfahrt eingesetzt zu werden.
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162 Mehr lesen ### Tensorial Curvature Measures in Integral Geometry

As pointed out in Section 3.1, there exist more tensor-valued generalizations of the curvature measures for convex bodies than the “obviously appearing” tensorial curvature measures. The generalized tensorial curvature measures φ r,s,1 j also admit a continuous extension to the convex bodies. In the upcoming (local) integral formulae, we will observe that these are not only a theoretical construct, but they appear naturally in the representation of kinematic and Crofton integrals of tensorial curvature measures for convex bodies. Since we aim to obtain integral formulae for Minkowski tensors by globalization of the corresponding local formulae, the valuations φ r,s,1 j (·, R n ) likewise occur. Although the generalized tensorial curvature measures have no global counterpart, in the same way as the Minkowski tensors are essentially the total tensorial curvature measures, Alesker’s characterization theorem shows that these are representable in terms of Minkowski tensors. The exact form of this representation has already been proved by McMullen (see [68, p. 269]). As it is an important tool in the upcoming proofs, we recall it in this section.
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164 Mehr lesen ### Das Riemannsche Integral (PDF, 54,7 KB)

Falls f eine Treppenfunktion ist, dann stimmen die Ober- und Unterintegrale von f ¨uberein... Weiters ist das integral monoton d.h..[r] ### Integral point sets over finite fields

end For further examples we refer to  where we list the coordinates of one extremal example for p ≤ 113. A formal proof of the correctness of the proposed algorithm is not difficult but a bit technical and so left to the reader. We remark that there are several non- isomorphic integral point sets in general position which achieve the upper bound ˙I(Z n , 2). So far we have no insight in their structure or in the asymptotic behavior ### Orthogonal polynomials and recurrence equations, operator equations and factorization

orthogonal polynomials, Hahn class, differential equations, difference equations, q-difference equations, hypergeometric functions, factorization of operator polynomials, computer algebr[r] ### Integral Field Spectroscopy Observations of SN 1987A

The first epoch from 2004 was observed as a part of the science verification run on SINFONI, and thus were the first data of this kind of SN 1987A. The second epoch was observed in 2005 after I was awarded time via a normal proposal process (Principal Investigator: K. Kjær, pro- posal id: 080.D-0588C). I planned the observations based on the Science verification, where I had established which setting of the instrument would be most beneficial for the scientific output. Integral field spectroscopy is also called 3D spectroscopy, since the data format is 3- dimensional (position on sky and wavelength) and best represented as a cube. While this data- format has been used for some years for galaxies, I had to develop new analysing techniques and considerations, because SN 1987A is an emission line object, where the lines originate from different physical processes. Chapter 2 lays out the observational technique behind the instrument, the instrument pipeline which constructs the data format, and the methodology I developed in order to interpret the data. Chapter 3 explains the data reduction and calibration of the two epochs, and Chapter 4 shows the spectra and line identification. In Chapter 5 I analyse the circumstellar ring in ways only possible with 3D spectroscopy. I have with our data been able to separate the ejecta emission of the supernova from the circumstellar ring, and I describe our findings in Chapter 6. I finally discuss and summarise our results in Chapter 7.
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120 Mehr lesen ### Coupled system of Maxwell equations and circuit equations in electro-magnetism

Co-simulation An approach to deal with dynamical simulation of the cou- pled systems is co-simulation [Sch11, SBG + 14], where each subsystem can be separately simulated in a specific simulation software package on its own time scale and most likely in different space domains. Then to simulate the full system, subsystems (i.e., Maxwell equations and circuit equations) are succes- sively solved where the repetition is done for each time step till convergence [MEJW14]. The used co-simulation technique in here is the serial scheme called Gauss-Seidel in contrast to the parallel scheme Jacobi. For instance, to approx- imate the Maxwell equations in space, we employ the C++ library Concepts in the frequency domain. Therefore, for the DAE analysis, we first use the stan- dard simulator for the Maxwell equations in the frequency domain and then transform the solution to the time domain to be entered in the circuit simula- tor. On the other hand, dynamical simulation of the circuit equations can be done by MATLAB software, where the time integration is carried out within the observation time interval I ∶= t k +1 − t k associated to each iteration k. The
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134 Mehr lesen ### New facts around the Choquet integral

ec dgfihjdgkml7naoapjqerts u'vtv dgwjdyzjx {}| fil~st7s.[r] ### Integral representations of projection functions [online]

volume of K|L, and L ranges over L d j , the Grassmann manifold of j-dimensional linear subspaces of R d . The only known integral representation is (1.2), i. e. for the case j = d − 1 (where L d d−1 and S d−1 are identified). However, if we consider only centrally symmetric convex bodies that fulfill a certain smoothness condition, there exists the integral representation ### Hauptsatz der Differential- und Integralrechnung und Folgerung für bestimmtes Integral

Teil2: Umkehrung gilt nicht, also nicht jede Stammfunktion ist Integralfunktion von f.[r] ### Aufgaben zum Integral der e-Funktion

3.3 Die Beliebtheit der Kandidaten kann durch den Wert des bestimmten Integrals ausgedrückt werden, wobei ein größerer Wert größere Beliebtheit anzeigt. Entscheiden Sie[r]