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.

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.

This open access two-volume set constitutes the proceedings of the 26th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2020, which took place in Dublin, Ireland, in April 2020, and was held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The total of 60 regular papers presented in these volumes was carefully reviewed and selected from 155 submissions. The papers are organized in topical sections as follows: Part I: Program verification; SAT and SMT; Timed and Dynamical Systems; Verifying Concurrent Systems; Probabilistic Systems; Model Checking and Reachability; and Timed and Probabilistic Systems. Part II: Bisimulation; Verification and Efficiency; Logic and Proof; Tools and Case Studies; Games and Automata; and SV-COMP 2020.

These conference proceedings present authoritative papers on new experience and research, particularly that which has led to advances in design procedures. It covers design, construction and performance experience of coastal structures and breakwaters particularly including new developments.