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In writing this book, the first problem I faced is to choose the contents of the book. Today the Smarandache Notions is so vast and diversified that it is indeed difficult to choose the materials. Then, I fixed on five topics, namely, Some Smarandache Sequences, Smarandache Determinant Sequences, the Smarandache Function and its generalizations, the Pseudo Smarandache Function and its generalizations, and Smarandache Number Related Triangles. Most of the results appeared before, but some results, particularyly in Chapter 5, are new. In writing this book, I took the freedom of including the more recent results, found by other researchers till 2009, to keep the expositions up to date. At the end of Chapter1, Chapter 3, Chapter4, and Chapter 5, some open problems are given in the form of conjectures and questions....

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Introduction 7-8 Chapter 1, Some Smarandache Sequences, 9-38 Chapter 2, Smarandache Determinant Sequences, 39-52 Chapter 3, The Smarandache Function, 53-88 Chapter 4, The Pseudo Smarandache Function, 89-168 Chapter 5, Smarandache Number Related Triangles, 169-192...

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The fourth volume, in my book series of “Collected Papers”, includes 100 published and unpublished articles, notes, (preliminary) drafts containing just ideas to be further investigated, scientific souvenirs, scientific blogs, project proposals, small experiments, solved and unsolved problems and conjectures, updated or alternative versions of previous papers, short or long humanistic essays, letters to the editors...

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This short technical paper advocates a bootstrapping algorithm from which we can form a statistically reliable opinion based on limited clinically observed data, regarding whether an osteo-hyperplasia could actually be a case of Ewing’s osteosarcoma. The basic premise underlying our methodology is that a primary bone tumour, if it is indeed Ewing’s osteosarcoma, cannot increase in volume beyond some critical limit without showing metastasis. We propose a statistical method to extrapolate such critical limit to primary tumour volume. Our model does not involve any physiological variables but rather is entirely based on time series observations of increase in primary tumour volume from the point of initial detection to the actual detection of metastases....

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Collected Eclectic Ideas - preface by the author.............................3 Contents....................................................6 ASTRONOMY..................................14 1. First Lunar Space Base, project proposal, by V. Christianto, Florentin Smarandache..15 2. On Recent Discovery of New Planetoids in the Solar System and Quantization of Celestial System, by V. Christianto, F. Smarandache..................28 3. Open and Solved Elementary Questions in Astronomy, by Florentin Smarandache.. 36 BIOLOGY......................................40 4. Statistical Modeling of Primary Ewing Tumors of the Bone, by Sreepurna Malakar, Florentin Smarandache, Sukanto Bhattacharya, in in , Vol. 3, No. JJ05, 81-88, 2005................41 CALCULUS....................................53 5. A Triple Inequality with Series and Improper Integrals, by Florentin Smarandache, in Bulletin of Pure and Applied Sciences, Vol. 25E, No. 1, 215-217, 2006.........54 6. Immediate Calculation of Some Poisson Type Integrals Using SuperMathematics Circular Ex-Centric Functions, by Florentin Smarandache & Mircea Eugen................................

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Florentin Smarandache, an American mathematician of Romanian descent has generated a vast variety of mathematical problems. Some problems are easy, others medium, but many are interesting or unsolved and this is the reason why the present book appears. Here, of course, there are problems from various types. Solving these problems is addictive like eating pumpkin seed: having once started, one cannot help doing it over and over again....

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Preface 5 Definitions, Problems, Theorems 7 References 76

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This book is a declaration of spiritual warfare, a call to rekindle the fire of the Reformation! Although this battle has been ongoing through many ages, it is time for this generation to know that it must chose which side it will fight on. There can be no peace between Christianity and the religion of the pope! Will you stand for the Reformation and for the gospel of Jesus Christ, or will you capitulate into apostasy?...

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We are saved by grace alone, not grace and the power of human free will; but by God’s grace alone! A grace of infinite and eternal power which always accomplishes what it sets out to do, and it never fails! The question is not just was the death of Christ necessary, but is it sufficient to save us from all our sins and completely set us free from the ancient curse? That is the fundamentally crucial and pivotal question that is constantly overlooked in current dialogues about Evangelicals and Roman Catholics together! Most people don’t even realize how different the papist and the biblical teaching on the atonement are! The Roman Catholic Church teaches that the merits of Christ’s life and death as well as the merits of all the saint s are now held in this thing called the treasury of merits. These merits must now be applied to us through human works and sacraments, which is so blasphemous because now, they teach that even the benefits of Christ’s sacrifice and his righteous life must be earned! The Roman Catholic Church teaches that the benefits of Christ’s sacrifice are now withheld from you in a reservoir called the treasury of m...

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Part 1: The Sufficiency of Christ as Our Perfect High Priest Part 2: The Sufficiency of Christ as Head of the Church

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The foundation of the doctrine of sola fide and sola gratia is none other than the finished sacrifice of Christ. This book is meant to challenge the Roman Catholic and indirectly even the Eastern Orthodox teachings concerning salvation. I hope that it will help Protestants defend their faith in an age when apostasy is rampant all around us....

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I hope you will come to agree with me that Jesus paid it all on the cross and now our salvation does not rest on our good works or our good deeds. As saint Paul says, “For if justification comes through the law, then Christ died for nothing” (Galatians 2:21). Jesus died entirely in our place and took upon himself the punishment the punishment we deserve. I love the way Cyril of Alexandria says it in the following words, "They led away the author of life to die – to die for our sake. In a way beyond our understanding, the power of God brought from Christ’s passion an end far different from that intended by his enemies. His suffering served as a snare for death and rendered it powerless. The Lord’s death proved to be our restoration to immortality and newness of life. Condemned to death though innocent, he went forward bearing on his shoulders the cross on which he was to suffer. He did this for our sake, taking on himself the punishment that the law justly imposed on sinners. He was cursed for our sake according to the saying of Scripture: “A curse is on everyone who is hanged on a tree.”" Then Cyril emphasizes his point even more b...

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-Introduction -The First Letter To Daniel Jodoin -The Second Letter To Daniel Jodoin -The Third Letter To Daniel Jodoin -The Fourth Letter To Daniel Jodoin -The Fifth Letter To Daniel Jodoin -The Sixth Letter To Daniel Jodoin -The Seventh Letter To Daniel Jodoin -The Eighth Letter To Daniel Jodoin...