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"This textbook presents a unified approach to compact and noncompact Riemann surfaces from the point of view of the so-called L2 $\bar{\delta}$-method. This method is a powerful technique from the theory of several complex variables, and provides for a unique approach to the fundamentally different characteristics of compact and noncompact riemann surfaces."

"The book develops the classical Chebyshev's approach which gives analytical representation for the solution in terms of Riemann surfaces. The techniques born in the remote (at the first glance) branches of mathematics such as complex analysis, Riemann surfaces and Teichmüller theory, foliations, braids, topology are applied to approximation problems. The key feature of this book is the usage of beautiful ideas of contemporary mathematics for the solution of applied problems and their effective numerical realization. This is one of the few books where the computational aspects of the higher genus Riemann surfaces are illuminated. Effective work with the moduli spaces of algebraic curves provides wide opportunities for numerical experiments in mathematics and theoretical physics.​"

"ABSTRACT: In this paper, the Riemann solution of an extended Riemann problem in channelnetworks is presented. The Riemann problem at a junction network is well defined in theliterature. However, it is limited to symmetric networks. Here, we extend the Riemann problemto non-symmetric networks such that neither the channel width equality nor the dischargeequality are assumed. The Riemann solution is given under subcritical flow conditions to ensurethe existence and uniqueness of the solution at the junction. Taking into account the massand energy conservation laws, the necessary conditions for the Riemann solution are drawn.The results are summarized in a theorem. The theorem is illustrated with a set of numericalexamples.In order to perform a one-dimensional simulation in channel networks, the inner boundaryconditions at the junction (i.e., the channel intersection point) are required. It has turned outthat the classical models (i.e., the Equality, Gurram, Hsu models) that have been used to supplysuch a boundary suffer from many drawbacks.Thus, here we propose to use the Riemann solution at the junction networks to provideproper boundary conditions. Then, we compare all the junction models together. The junctionmodels are validated against experimental results found in the literature for steady state flows.Generally, the Riemann model shows good results in matching the experimental data. Inparticular, the Riemann model shows the best results when the bottom discontinues at thejunction. For the unsteady state flows, we perform prototype case studies to test the junctionmodels in the channel networks, and the numerical solutions are compared with the analyticalsolutions. The Riemann model continues to show the best results that agree with the analyticalsolutions. However, the validation of the junction models in the unsteady state flows remainsfor future work due to the limited amount of real data."

"Integral Henstock-Kurzweil merupakan hasil dari perkembangan integral Riemann. Dalam tulisan ini akan ditunjukkan bagaimana sifat-sifat integral Riemann dan Integral Henstock-Kurzweil dari fungsi bernilai di ruang Banach. Selain itu, akan ditunjukkan perbandingan antara integral Riemann dan integral Henstock-Kurzweil untuk fungsi bernilai di ruang Banach berdimensi takhingga.

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Henstock-Kurzweil integrable is Generalized Riemann integrable. In this paper, will show the property of Riemann integrable and Henstockk-Kurzweil integrable of function Banach-valued. And comparison Riemann integrable and Henstock-Kurzweil integrable for infinite Banach space."

"Integral Riemann-Stieltjes merupakan bentuk yang lebih umum dari integral Riemann. Integral Riemann-Stieltjes berbobot adalah integral Riemann-Stieltjes yang melibatkan perkalian dua fungsi pada integran. Salah satu metode untuk mengaproksimasi integral Riemann-Stieltjes berbobot adalah aturan trapesium berbobot. Namun, terdapat galat pada aproksimasi tersebut ketika menggunakan aturan tersebut. Studi literatur ini bertujuan untuk mempelajari batas galat pada aproksimasi integral Riemann-Stieltjes dengan menggunakan aturan trapesium berbobot. Fungsi-fungsi yang digunakan pada integran dan integrator adalah fungsi kontinu, fungsi monoton, fungsi Lipschitz, dan fungsi variasi terbatas.

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Riemann-Stieltjes integral is a generalization of the Riemann integral. Weighted Riemann-Stieltjes integral is a Riemann-Stieltjes integral which involves product of two functions. One of many methods to approximate weighted Riemann-Stieltjes integral is weighted trapezoudal rule. However, there is an error in approximating the value by using this method. The focus of this study is the error bounds in approximating the weighted Riemann-Stieltjes integral by the weighted trapezoidal rule. Classes of functions such as functions of bounded variation, continuous, monotonic, and Lipschitzian functions are the integrands and integrators that are discussed."

"The book provides an introduction to differential geometry of curves and surfaces. The theory of curves starts with a discussion of possible definitions of the concept of curve, proving in particular the classification of 1-dimensional manifolds. We then present the classical local theory of parametrized plane and space curves (curves in n-dimensional space are discussed in the complementary material): curvature, torsion, Frenet’s formulas and the fundamental theorem of the local theory of curves. Then, after a self-contained presentation of degree theory for continuous self-maps of the circumference, we study the global theory of plane curves, introducing winding and rotation numbers, and proving the Jordan curve theorem for curves of class C2, and Hopf theorem on the rotation number of closed simple curves. The local theory of surfaces begins with a comparison of the concept of parametrized (i.e., immersed) surface with the concept of regular (i.e., embedded) surface. We then develop the basic differential geometry of surfaces in R3: definitions, examples, differentiable maps and functions, tangent vectors (presented both as vectors tangent to curves in the surface and as derivations on germs of differentiable functions; we shall consistently use both approaches in the whole book) and orientation. Next we study the several notions of curvature on a surface, stressing both the geometrical meaning of the objects introduced and the algebraic/analytical methods needed to study them via the Gauss map, up to the proof of Gauss’ Teorema Egregium. Then we introduce vector fields on a surface (flow, first integrals, integral curves) and geodesics (definition, basic properties, geodesic curvature, and, in the complementary material, a full proof of minimizing properties of geodesics and of the Hopf-Rinow theorem for surfaces). Then we shall present a proof of the celebrated Gauss-Bonnet theorem, both in its local and in its global form, using basic properties (fully proved in the complementary material) of triangulations of surfaces. As an application, we shall prove the Poincaré-Hopf theorem on zeroes of vector fields. Finally, the last chapter will be devoted to several important results on the global theory of surfaces, like for instance the characterization of surfaces with constant Gaussian curvature, and the orientability of compact surfaces in R3."

"Subdivision surfaces permit a designer to specify the approximate form of a surface defining an object and to refine and smooth the form to obtain a more useful or attractive version of the surface."