The Concrete Tetrahedron

The book treats four mathematical concepts which play a fundamental role in many different areas of mathematics: symbolic sums, recurrence (difference) equations, generating functions, and asymptotic estimates. Their key features, in isolation or in combination, their mastery by paper and pencil or by computer programs, and their applications to problems in pure mathematics or to "real world problems" (e.g. the analysis of algorithms) are studied. The book is intended as an algorithmic supplement to the bestselling "Concrete Mathematics" by Graham, Knuth and Patashnik.

Through sheer determination and courage, Kahn has researched the nature of concrete in the form of precast, cast in place or blocks. Each of his renowned works in exposed concrete, such as the Yale Art Gallery, the Richards Laboratories, the Bath House, the Salk Institute, the National Assembly, the Kimbell Museum, the Exeter Library and the Yale Center for British Art, is itself an important chapter in the history of architecture for the exploration into concrete’s formal expression, beyond the lesson of Le Corbusier. Kahn’s obsession on concrete fabrication processes, on the formwork and the mix design, is systematically examined in two volumes. The authors illustrate Kahn’s vision with documents that have never been revealed in other essays, drawing heavily from original sketches, plans, specifications, worksite photographs, and correspondences with collaborators, engineers, technicians and contractors. The first volume Exposed Concrete and Hollow Stones focuses on the first ten-year period of Kahn’s research on concrete. Moving through the many construction systems experienced by Kahn, from the discovery of exposed concrete in the form of béton brut at the Yale Art Gallery, to the precast and poured-in-place techniques, to the values of joint, growth and ornament, the essay culminates in the reconstruction of the artistic and technical characteristics of two great worksite, the Richards Laboratories and the First Unitarian Church and School. The second volume, Towards the Zero Degree of Concrete, covers the following fourteen years and leads the reader along Kahn’s path to the true “nature of concrete,” focusing on his main techniques and poetic discoveries such as the “liquid stone” of the Salk Institute, the “smooth finish” at Bryn Mawr and the concept of “monolithic” at the Yale Center for British Art.

The diversity of constructions included in this publication on space structures ranges from anten reflectors and masts positioned in space, to equally exciting terrestrial structures, notably large-span domes, barrel vaults, multi-layered grids, cable and membrane systems, and pneumatic structures. This collection of more than two hundred and twenty papers, presented in to volumes, is the work of leading international experts for presentation at the Fourth International Conference on Space Structures. These two volumes contain a prodigious amount of original and innovative information on space structures that will be of especial interest to engineers, architects and other professionals engaged in the planning, design, fabrication and erection of novel constructions.

As both chemist and priest your scribe believes that the current gap between science and religion can be bridged largely by revelation. Revelation is a select part of religion, often beyond the ken or competence of qualified science. Types of revelation comprise the manifest supernatural and prophecy, fulfilled prophecy supporting what is yet to be fulfilled. The book offers answers and asks a variety of questions. This book is written in four sections, each with chapter-like and numbered subsections: Section 1 the most scriptural and salvational Section 2 the most prophetic or revelatory Section 3 the most scientific and integrative knowledge Section 4 the most semantic and hypothetic Section 1 Scripture, Old and New Testament, appears to be a rich source of revelation and other reliable spiritual reality. Its integrity distinguishes divine and human reporting, also religion versus irreligion. Jesus' early advent fulfilled dozens of Old Testament prophecies; divine evidence for the reliability of its revelation. Scripture reveals that Jesus of Nazareth walked among us, both man and God. Section 2 Section 2 comprise a commentary upon the Revelation to John. The prophecy concentrated therein is mysterious in part yet relatively ordered and culminating. It helps to organize other prophecy revealed in Scripture. And it serves to guide our on-going participation with the ascended Christ as Lord. Prophecy reveals that God has operated mightily in and on history, that he has revealed essential parts of his plan and care for mankind. Section 3 Without religion, science, particularly inanimate science, tends to support determinism, also a relatively rigid causation or rationalism. Science develops knowledge more than understanding. Section 3 attempts to assemble salient science together with a minor proportion of related hypotheses. Your scribe believes that God's concern and involvement and control of life is more intimate and profound than most science and philosophy has indicated. Section 4 The relatively hypothetic Section 4 comprises much supposition, some semantically treated. Suppositions are offered concerning material or systematic structures for said living sub matter in body, mind and soul. Life after first death is a gift from the soul's Creator. Spirits just and unjust await resurrection in the spirit, not in the flesh, not in reincarnation. Tthe soul is foundational to theology and tends to respond to spiritual reality, to living sub matter, particularly to God and other souls.

Defined as solutions of linear differential or difference equations with polynomial coefficients, D-finite functions play an important role in various areas of mathematics. This book is a comprehensive introduction to the theory of these functions with a special emphasis on computer algebra algorithms for computing with them: algorithms for detecting relations from given data, for evaluating D-finite functions, for executing closure properties, for obtaining various kinds of “explicit” expressions, for factoring operators, and for definite and indefinite symbolic summation and integration are explained in detail. The book comes “with batteries included” in the sense that it requires no background in computer algebra as the relevant facts from this area are summarized in the beginning. This makes the book accessible to a wide range of readers, from mathematics students who plan to work themselves on D-finite functions to researchers who want to apply the theory to their own work. Hundreds of exercises invite the reader to apply the techniques in the book and explore further aspects of the theory on their own. Solutions to all exercises are given in the appendix. When algorithms for D-finite functions came up in the early 1990s, computer proofs were met with a certain skepticism. Fortunately, these times are over and computer algebra has become a standard tool for many mathematicians. Yet, this powerful machinery is still not as widely known as it deserves. This book helps to spread the word that certain tasks can be safely delegated to a computer algebra system, and also what the limitations of these techniques are.