How to execute repeated integral to generate hyper-exponential functions

As mentioned above, we need repeated integral to generate hyperexponential functions of n-order. By the way, what is the way how to execute repeated integral by using a computer?

Computation of Definite Integral Over Repeated Integral

The tasks involving repeated integral occur from time to time in technical practice. This paper introduces the research of authors in the field of repeated integrals within the required class of functions. Authors focus on the definite integral over repeated integral and they develop a tool for its computation. It involves two principal steps, analytical and numerical step. In the analytical step, the definite integral over a repeated integral is decomposed into n integrals and then the Cauchy form is used for further rearrangement. Numerical step involves Gauss type integration slightly modified by the authors. Several examples illustrating the operation of both analytical and numerical steps of the method are provided in the paper.

DEVELOPING E-SCAFFOLDING TO IMPROVE THE QUALITY OF PROCESS AND LEARNING OUTCOMES

PENGEMBANGAN E-SCAFFOLDING UNTUKMENINGKATKAN KUALITAS PROSES DAN HASIL BELAJARAbstrakPenelitian ini bertujuan untuk mengembangkan produk E-Scaffolding untuk meningkatkan kualitas proses dan hasil belajar mahasiswa pada Mata Kuliah Fisika Matematika. mata kuliah fisika yang menjadi pokok bahasan pada penelitian ini adalah Fisika matematika khususnya pada pokok bahasan integral lipat. Metode penelitian yang digunakan untuk mengembangkan e-scaffolding adalah Research and Development. Desain tersebut meliputi 4 tahapan, yaitu analisis kebutuhan, desain model , pengembangan model, validasi, revisi dan uji coba produk Instrumen penelitian yang digunakan adalah lembar validasi dari ahli materi dan ahli media, angket, dan tes. Hasil penelitian menghasilkan sebuah e-scaffolding, yakni produk pendukung pembelajaran online dengan menggunakan website dan fasilitas scaffolding. Hasil uji kelayakan menunjukan menunjukkan nilai yang sangat baik dari ahli media dan sangat layak dari ahli materi. Penggunaan e-scaffolding memang dapat meningkatkan hasil belajar siswa dan kualitas pembelajaran di kelas menjadi lebih efektif dan efisien. Hasil analisis kualitas proses pembelajaran dengan menggunakan e-scaffolding juga menunjukkan adanya peningkatan dibandingkan dengan menggunakan pembelajaran direct instruction.Kata kunci: E-scaffolding, hasil belajar, Fisika MatematikaAbstractThis study was aimed at developing E-scaffolding to improve the quality of the process and product of students’ learning of Physical Methematics. The physics subject of this study was Physical Mathematics focusing on the topic of repeated integral. The method used to develop the E-scaffolding was Research and Development. The design covered 4 stages, namely need analysis, model design, model development and validation, and revision and product testing. The research instruments used in this study were validation sheets from material and media experts, questionnaires, and tests. The results of the study generates an E-scaffolding, which is a product of online learning aids using website and scaffolding facilities. The feasibility test results show that the E-scaffolding is categorized into the very good category tested by the media experts and the highly feasible category tested by the material experts. The use of E-scaffolding is able to improve the students’ learning outcomes and the quality of the classroom teaching learning process. The results of the quality analysis of the learning process using E-scaffolding also shows an improvement compared to the use of the direct instruction method.Keywords: E-scaffolding, learning outcome, Physical Mathematics

Consolidation under embankment-type loads

Plane strain problems of consolidation (or poro-elasticity) can be solved using the two displacement functions defined by McNamee and Gibson with the help of a repeated integral transformation technique. The problem of a semi-infinite clay layer whose surface is subjected to an embankment-type of normal trapezoidal pressure applied along an infinite strip is treated here. The general loading pattern selected easily degenerates into a rectangular (uniformly distributed) load for which NcNamee and Gibson gave the solutions, to the triangular loads and also to the line loads. Not only the settlements, but also the pore pressures have been evaluated under these types of loads when the surface is either pervious or impervious.The nondimensional solutions presented are useful to highway and embankment engineers. There is also an example of the use of these solutions.

A uniqueness theorem for the exponential series of Herglotz

If {λ;n}, {bn} are sequences of complex numbers, and we consider the series ∑bn exp (−λnx), given as convergent in (0, 1) (i.e. the open invertal (0,1)) to f(x)∈L, then, writing(if λn = 0 the corresponding term is ½bnx2) where the series is supposed is to be uniformly convergent in (0, 1), we havefor 0<h<h(x).If we know that the second member of (2) tends to f(x) as h → +0, it will follow that F(x) is a repeated integral of f(x) ((1), 671). If there is a sequence {φv(x)} of integrable functions with the property thatthen, on multiplying (1) by φv(x) and integrating over (0,1), we obtain a formula for bv in terms of F(x). On integrating by parts twice, bv will be expressed in terms of f(x), and this will constitute a uniqueness theorem for the series ∑bn exp (−λnx).

The evaluation of matrix elements in functional integral form

The source-function technique of Schwinger is used to express any transformation matrix element in a form involving a functional derivative operator. An elementary algebra of such expressions is developed, and in the case of quantum mechanics is found to lead to a natural evaluation procedure; the result expresses the matrix element as an indefinitely repeated integral, with the integrations arranged in order of importance. Discrete as well as continuous eigenvalues can be dealt with by this method. In so far as the procedure is a kind of ‘sum-over-histories’, the result is related to the Feynman principle. The field-theoretic situation requires further developments in technique.