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| the tools of mathematical reasoning lakins solutions: The Tools of Mathematical Reasoning Tamara J. Lakins, 2016-09-08 This accessible textbook gives beginning undergraduate mathematics students a first exposure to introductory logic, proofs, sets, functions, number theory, relations, finite and infinite sets, and the foundations of analysis. The book provides students with a quick path to writing proofs and a practical collection of tools that they can use in later mathematics courses such as abstract algebra and analysis. The importance of the logical structure of a mathematical statement as a framework for finding a proof of that statement, and the proper use of variables, is an early and consistent theme used throughout the book. |
| the tools of mathematical reasoning lakins solutions: Checking for Understanding Douglas Fisher, Nancy Frey, 2015-12-18 A teacher presents a lesson, and at the end asks students if they understand the material. The students nod and say they get it. Later, the teacher is dismayed when many of the students fail a test on the material. Why aren’t students getting it? And, just as important, why didn’t the teacher recognize the problem? In Checking for Understanding, Douglas Fisher and Nancy Frey show how to increase students’ understanding with the help of creative formative assessments. When used regularly, formative assessments enable every teacher to determine what students know and what they still need to learn. Fisher and Frey explore a variety of engaging activities that check for and increase understanding, including interactive writing, portfolios, multimedia presentations, audience response systems, and much more. This new 2nd edition of Checking for Understanding has been updated to reflect the latest thinking in formative assessment and to show how the concepts apply in the context of Fisher and Frey’s work on gradual release of responsibility, guided instruction, formative assessment systems, data analysis, and quality instruction. Douglas Fisher and Nancy Frey are the creators of the Framework for Intentional and Targeted (FIT) Teaching™. They are also the authors of numerous ASCD books, including The Formative Assessment Action Plan: Practical Steps to More Successful Teaching and Learning and the best-selling Enhancing RTI: How to Ensure Success with Effective Classroom Instruction and Intervention. |
| the tools of mathematical reasoning lakins solutions: Tools for Teaching Logic Patrick Blackburn, Hans van Ditmarsch, Maria Manzano, Fernando Soler-Toscano, 2011-05-23 This book constitutes the proceedings of the Third International Congress on Tools for Teaching Logic, TICTTL 2011, held in Salamanca, Spain, in June 2011. The 30 papers presented were carefully reviewed and selected from 62 submissions. The congress focusses on a variety of topics including: logic teaching software, teaching formal methods, logic in the humanities, dissemination of logic courseware and logic textbooks, methods for teaching logic at different levels of instruction, presentation of postgraduate programs in logic, e-learning, logic games, teaching argumentation theory and informal logic, and pedagogy of logic. |
| the tools of mathematical reasoning lakins solutions: The Blue Book of Grammar and Punctuation Lester Kaufman, Jane Straus, 2021-04-16 The bestselling workbook and grammar guide, revised and updated! Hailed as one of the best books around for teaching grammar, The Blue Book of Grammar and Punctuation includes easy-to-understand rules, abundant examples, dozens of reproducible quizzes, and pre- and post-tests to help teach grammar to middle and high schoolers, college students, ESL students, homeschoolers, and more. This concise, entertaining workbook makes learning English grammar and usage simple and fun. This updated 12th edition reflects the latest updates to English usage and grammar, and includes answers to all reproducible quizzes to facilitate self-assessment and learning. Clear and concise, with easy-to-follow explanations, offering just the facts on English grammar, punctuation, and usage Fully updated to reflect the latest rules, along with even more quizzes and pre- and post-tests to help teach grammar Ideal for students from seventh grade through adulthood in the US and abroad For anyone who wants to understand the major rules and subtle guidelines of English grammar and usage, The Blue Book of Grammar and Punctuation offers comprehensive, straightforward instruction. |
| the tools of mathematical reasoning lakins solutions: A Handbook for the Art and Science of Teaching Robert J. Marzano, John L. Brown, 2009 A series of modules designed to help educators explore and put into practice the research findings presented in The Art and Science of Teaching. |
| the tools of mathematical reasoning lakins solutions: Modeling of Biological Materials Francesco Mollica, Luigi Preziosi, K. R. Rajagopal, 2007-12-31 This unique collection highlights the central role played by modeling in general, and the modeling of mechanical considerations that have an effect on living matter. The volume collects several survey papers by actively working specialists, dealing with some of the most important problems – both theoretical and practical – in biomechanics. Written in a user-friendly style, these papers clearly explain both the biomedical and mechanical backgrounds associated with complex phenomena. This book may be used in interdisciplinary introductory courses covering various biomechanical topics for graduate students in applied mathematics, engineering, and biomedicine. |
| the tools of mathematical reasoning lakins solutions: Complex Analysis with Applications Nakhlé H. Asmar, Loukas Grafakos, 2018-10-12 This textbook is intended for a one semester course in complex analysis for upper level undergraduates in mathematics. Applications, primary motivations for this text, are presented hand-in-hand with theory enabling this text to serve well in courses for students in engineering or applied sciences. The overall aim in designing this text is to accommodate students of different mathematical backgrounds and to achieve a balance between presentations of rigorous mathematical proofs and applications. The text is adapted to enable maximum flexibility to instructors and to students who may also choose to progress through the material outside of coursework. Detailed examples may be covered in one course, giving the instructor the option to choose those that are best suited for discussion. Examples showcase a variety of problems with completely worked out solutions, assisting students in working through the exercises. The numerous exercises vary in difficulty from simple applications of formulas to more advanced project-type problems. Detailed hints accompany the more challenging problems. Multi-part exercises may be assigned to individual students, to groups as projects, or serve as further illustrations for the instructor. Widely used graphics clarify both concrete and abstract concepts, helping students visualize the proofs of many results. Freely accessible solutions to every-other-odd exercise are posted to the book’s Springer website. Additional solutions for instructors’ use may be obtained by contacting the authors directly. |
| the tools of mathematical reasoning lakins solutions: Calculus for the Life Sciences Frederick R. Adler, Miroslav Lovric, |
| the tools of mathematical reasoning lakins solutions: Spaces Tom Lindstrøm, 2017 Spaces is a modern introduction to real analysis at the advanced undergraduate level. It is forward-looking in the sense that it first and foremost aims to provide students with the concepts and techniques they need in order to follow more advanced courses in mathematical analysis and neighboring fields. The only prerequisites are a solid understanding of calculus and linear algebra. Two introductory chapters will help students with the transition from computation-based calculus to theory-based analysis. The main topics covered are metric spaces, spaces of continuous functions, normed spaces, differentiation in normed spaces, measure and integration theory, and Fourier series. Although some of the topics are more advanced than what is usually found in books of this level, care is taken to present the material in a way that is suitable for the intended audience: concepts are carefully introduced and motivated, and proofs are presented in full detail. Applications to differential equations and Fourier analysis are used to illustrate the power of the theory, and exercises of all levels from routine to real challenges help students develop their skills and understanding. The text has been tested in classes at the University of Oslo over a number of years |
| the tools of mathematical reasoning lakins solutions: A Glossary of Words Used in Swaledale, Yorkshire John Harland, 1873 |
| the tools of mathematical reasoning lakins solutions: Proofs from THE BOOK Martin Aigner, Günter M. Ziegler, 2013-04-17 The (mathematical) heroes of this book are perfect proofs: brilliant ideas, clever connections and wonderful observations that bring new insight and surprising perspectives on basic and challenging problems from Number Theory, Geometry, Analysis, Combinatorics, and Graph Theory. Thirty beautiful examples are presented here. They are candidates for The Book in which God records the perfect proofs - according to the late Paul Erdös, who himself suggested many of the topics in this collection. The result is a book which will be fun for everybody with an interest in mathematics, requiring only a very modest (undergraduate) mathematical background. For this revised and expanded second edition several chapters have been revised and expanded, and three new chapters have been added. |
| the tools of mathematical reasoning lakins solutions: History of Fayette County, Illinois Brink, McDonough and Company, 1878 |
| the tools of mathematical reasoning lakins solutions: Fundamentals of Matrix Computations David S. Watkins, 1991-01-16 The use of numerical methods continues to expand rapidly. At their heart lie matrix computations. Written in a clear, expository style, it allows students and professionals to build confidence in themselves by putting the theory behind matrix computations into practice instantly. Algorithms that allow students to work examples and write programs introduce each chapter. The book then moves on to discuss more complicated theoretical material. Using a step-by-step approach, it introduces mathematical material only as it is needed. Exercises range from routine computations and verifications to extensive programming projects and challenging proofs. |
| the tools of mathematical reasoning lakins solutions: Partial Differential Equations with Fourier Series and Boundary Value Problems Nakhle H. Asmar, 2017-03-23 Rich in proofs, examples, and exercises, this widely adopted text emphasizes physics and engineering applications. The Student Solutions Manual can be downloaded free from Dover's site; instructions for obtaining the Instructor Solutions Manual is included in the book. 2004 edition, with minor revisions. |
| the tools of mathematical reasoning lakins solutions: Instructional Planning for Effective Teaching James H. Stronge, Xianxuan Xu, 2015-11-13 Cultivate meaningful learning schoolwide. Taking a practical approach to instructional planning, the authors outline research-based planning tools and illustrate how teachers, leaders, and administrators can use these tools in everyday practice. Discover powerful strategies and guidelines for developing quality lessons, setting learning objectives, planning differentiated instruction, and designing technology-integrated learning to effectively teach and challenge every student. |
| the tools of mathematical reasoning lakins solutions: Elementary Set Theory, Part I K.T. Leung, Doris Lai-chue Chen, 1992-10-01 This book provides students of mathematics with the minimum amount of knowledge in logic and set theory needed for a profitable continuation of their studies. There is a chapter on statement calculus, followed by eight chapters on set theory. |
| the tools of mathematical reasoning lakins solutions: Elements of Set Theory Herbert B. Enderton, 1977-04-28 This is an introductory undergraduate textbook in set theory. In mathematics these days, essentially everything is a set. Some knowledge of set theory is necessary part of the background everyone needs for further study of mathematics. It is also possible to study set theory for its own interest--it is a subject with intruiging results anout simple objects. This book starts with material that nobody can do without. There is no end to what can be learned of set theory, but here is a beginning. |
| the tools of mathematical reasoning lakins solutions: The Differentiated School Carol Tomlinson, Kay Brimijoin, Lane Narvaez, 2008-06-15 Three educators share powerful, real-life accounts of how two schools successfully built differentiated instruction into every classroom and achieved sweeping positive results for their staff and students. |
| the tools of mathematical reasoning lakins solutions: The Music of Life Denis Noble, 2008-02-14 What is Life? Decades of research have resulted in the full mapping of the human genome - three billion pairs of code whose functions are only now being understood. The gene's eye view of life, advocated by evolutionary biology, sees living bodies as mere vehicles for the replication of the genetic codes.But for a physiologist, working with the living organism, the view is a very different one. Denis Noble is a world renowned physiologist, and sets out an alternative view to the question - one that becomes deeply significant in terms of the living, breathing organism. The genome is not life itself. Noble argues that far from genes building organisms, they should be seen as prisoners of the organism.The view of life presented in this little, modern, post-genome project reflection on the nature of life, is that of the systems biologist: to understand what life is, we must view it at a variety of different levels, all interacting with each other in a complex web. It is that emergent web, full of feedback between levels, from the gene to the wider environment, that is life. It is a kind of music.Including stories from Noble's own research experience, his work on the heartbeat, musical metaphors, and elements of linguistics and Chinese culture, this very personal and at times deeply lyrical book sets out thesystems biology view of life. |
| the tools of mathematical reasoning lakins solutions: The Handbook for the New Art and Science of Teaching Robert J. Marzano, 2018-09-21 Companion volume to The new art and science of teaching. |
| the tools of mathematical reasoning lakins solutions: Employment-unemployment United States. Congress. Joint Economic Committee, 1990 |
| the tools of mathematical reasoning lakins solutions: Single-Cell-Based Models in Biology and Medicine Alexander Anderson, Katarzyna Rejniak, 2007-08-07 Many different single-cell-based models have been developed and applied to biological and medical problems. Computational approaches used are Monte-Carlo simulations, energy minimisation techniques, volume conservation laws, solutions of the equations of motion for each individual cell or for each point on the cell membrane. They differ in the level of detail that defines the cell structure and subsequently in the number of individual cells that the model can incorporate. This volume presents a collection of mathematical and computational single-cell-based models and their application. The main sections cover four general model groupings: hybrid cellular automata, cellular potts, lattice-free cells, and viscoelastic cells. Each section is introduced by a discussion of the applicability of the particular modelling approach and its advantages and disadvantages, which will make the book suitable for students starting research in mathematical biology as well as scientists modelling multicellular processes. |
| the tools of mathematical reasoning lakins solutions: Philosophy of Cancer Marta Bertolaso, 2016-08-24 Since the 1970s, the origin of cancer is being explored from the point of view of the Somatic Mutation Theory (SMT), focusing on genetic mutations and clonal expansion of somatic cells. As cancer research expanded in several directions, the dominant focus on cells remained steady, but the classes of genes and the kinds of extra-genetic factors that were shown to have causal relevance in the onset of cancer multiplied. The wild heterogeneity of cancer-related mutations and phenotypes, along with the increasing complication of models, led to an oscillation between the hectic search of ‘the’ few key factors that cause cancer and the discouragement in face of a seeming ‘endless complexity’. To tame this complexity, cancer research started to avail itself of the tools that were being developed by Systems Biology. At the same time, anti-reductionist voices began claiming that cancer research was stuck in a sterile research paradigm. This alternative discourse even gave birth to an alternative theory: the Tissue Organization Field Theory (TOFT). A deeper philosophical analysis shows limits and possibilities of reductionist and anti-reductionist positions and of their polarization. This book demonstrates that a radical philosophical reflection is necessary to drive cancer research out of its impasses. At the very least, this will be a reflection on the assumptions of different kinds of cancer research, on the implications of what cancer research has been discovering over 40 years and more, on a view of scientific practice that is most able to make sense of the cognitive and social conflicts that are seen in the scientific community (and in its results), and, finally, on the nature of living entities with which we entertain this fascinating epistemological dance that we call scientific research. The proposed Dynamic and Relational View of carcinogenesis is a starting point in all these directions. |
| the tools of mathematical reasoning lakins solutions: Discrete Convex Analysis Kazuo Murota, 2003-01-01 Discrete Convex Analysis is a novel paradigm for discrete optimization that combines the ideas in continuous optimization (convex analysis) and combinatorial optimization (matroid/submodular function theory) to establish a unified theoretical framework for nonlinear discrete optimization. The study of this theory is expanding with the development of efficient algorithms and applications to a number of diverse disciplines like matrix theory, operations research, and economics. This self-contained book is designed to provide a novel insight into optimization on discrete structures and should reveal unexpected links among different disciplines. It is the first and only English-language monograph on the theory and applications of discrete convex analysis. |
| the tools of mathematical reasoning lakins solutions: Mathematical Writing Donald E. Knuth, Tracy Larrabee, Paul M. Roberts, 1989 This book will help those wishing to teach a course in technical writing, or who wish to write themselves. |
| the tools of mathematical reasoning lakins solutions: Chain of Command Caspar Weinberger, Peter Schweizer, 2006-09-26 Secret Service agent Mike Delaney goes up against a ruthless hidden enemy with the cold-blooded will to assassinate the president of the United States--and frame Delaney for the murder. |
| the tools of mathematical reasoning lakins solutions: A Primer of Real Analytic Functions KRANTZ, PARKS, 2013-03-09 The subject of real analytic functions is one of the oldest in mathe matical analysis. Today it is encountered early in ones mathematical training: the first taste usually comes in calculus. While most work ing mathematicians use real analytic functions from time to time in their work, the vast lore of real analytic functions remains obscure and buried in the literature. It is remarkable that the most accessible treatment of Puiseux's theorem is in Lefschetz's quite old Algebraic Geometry, that the clearest discussion of resolution of singularities for real analytic manifolds is in a book review by Michael Atiyah, that there is no comprehensive discussion in print of the embedding prob lem for real analytic manifolds. We have had occasion in our collaborative research to become ac quainted with both the history and the scope of the theory of real analytic functions. It seems both appropriate and timely for us to gather together this information in a single volume. The material presented here is of three kinds. The elementary topics, covered in Chapter 1, are presented in great detail. Even results like a real ana lytic inverse function theorem are difficult to find in the literature, and we take pains here to present such topics carefully. Topics of middling difficulty, such as separate real analyticity, Puiseux series, the FBI transform, and related ideas (Chapters 2-4), are covered thoroughly but rather more briskly. |
| the tools of mathematical reasoning lakins solutions: Proceedings of the Casualty Actuarial Society Casualty Actuarial Society, 1999 List of members for the years 1914-20 are included in v. 1-7, after which they are continued in the Year book of the society, begun in 1922. |
| the tools of mathematical reasoning lakins solutions: Hollywood Highbrow Shyon Baumann, 2018-06-05 Today's moviegoers and critics generally consider some Hollywood products--even some blockbusters--to be legitimate works of art. But during the first half century of motion pictures very few Americans would have thought to call an American movie art. Up through the 1950s, American movies were regarded as a form of popular, even lower-class, entertainment. By the 1960s and 1970s, however, viewers were regularly judging Hollywood films by artistic criteria previously applied only to high art forms. In Hollywood Highbrow, Shyon Baumann for the first time tells how social and cultural forces radically changed the public's perceptions of American movies just as those forces were radically changing the movies themselves. The development in the United States of an appreciation of film as an art was, Baumann shows, the product of large changes in Hollywood and American society as a whole. With the postwar rise of television, American movie audiences shrank dramatically and Hollywood responded by appealing to richer and more educated viewers. Around the same time, European ideas about the director as artist, an easing of censorship, and the development of art-house cinemas, film festivals, and the academic field of film studies encouraged the idea that some American movies--and not just European ones--deserved to be considered art. |
| the tools of mathematical reasoning lakins solutions: Abstract Algebra Thomas W Judson, 2019-08 |
| the tools of mathematical reasoning lakins solutions: Vestnik Leningradskiĭ gosudarstvennyĭ universitet imeni A.A. Zhdanova, 1981 |
| the tools of mathematical reasoning lakins solutions: Quality Questioning Jackie Acree Walsh, Beth Dankert Sattes, 2004-10-27 Use effective questions to advance student thinking, learning, and achievement! Authors Walsh and Sattes provide an in-depth look at how quality questions can transform classrooms. Drawing on two decades of research on teacher effectiveness, the authors offer strategies that engage all students in the teacher’s questions and prompt students to generate their own questions. Quality Questioning includes: A complete framework for preparing and presenting questions, prompting and processing student responses, teaching students to generate questions, and reflecting on questioning practice Checklists for classroom applications Reproducibles, rubrics, resources, evaluation tools, and more |
| the tools of mathematical reasoning lakins solutions: A Beginner's Guide to Mathematical Logic Raymond M. Smullyan, 2014-07-23 Written by a creative master of mathematical logic, this introductory text combines stories of great philosophers, quotations, and riddles with the fundamentals of mathematical logic. Author Raymond Smullyan offers clear, incremental presentations of difficult logic concepts. He highlights each subject with inventive explanations and unique problems. Smullyan's accessible narrative provides memorable examples of concepts related to proofs, propositional logic and first-order logic, incompleteness theorems, and incompleteness proofs. Additional topics include undecidability, combinatoric logic, and recursion theory. Suitable for undergraduate and graduate courses, this book will also amuse and enlighten mathematically minded readers. Dover (2014) original publication. See every Dover book in print at www.doverpublications.com |
| the tools of mathematical reasoning lakins solutions: Classroom Assessment & Grading that Work Robert J. Marzano, 2006 Robert J. Marzano distills 35 years of research to bring you expert advice on the best practices for assessing and grading the work done by today's students. |
| the tools of mathematical reasoning lakins solutions: Classical Topics in Complex Function Theory Reinhold Remmert, 2013-03-14 An ideal text for an advanced course in the theory of complex functions, this book leads readers to experience function theory personally and to participate in the work of the creative mathematician. The author includes numerous glimpses of the function theory of several complex variables, which illustrate how autonomous this discipline has become. In addition to standard topics, readers will find Eisenstein's proof of Euler's product formula for the sine function; Wielandts uniqueness theorem for the gamma function; Stirlings formula; Isssas theorem; Besses proof that all domains in C are domains of holomorphy; Wedderburns lemma and the ideal theory of rings of holomorphic functions; Estermanns proofs of the overconvergence theorem and Blochs theorem; a holomorphic imbedding of the unit disc in C3; and Gausss expert opinion on Riemanns dissertation. Remmert elegantly presents the material in short clear sections, with compact proofs and historical comments interwoven throughout the text. The abundance of examples, exercises, and historical remarks, as well as the extensive bibliography, combine to make an invaluable source for students and teachers alike |
| the tools of mathematical reasoning lakins solutions: The Florida Cracker Cookbook Joy Sheffield Harris, 2019-10-28 This Florida Book Awards Gold Medal-winner in the Cooking category celebrates the Sunshine State’s culinary heritage—from turtle soup to boiled peanuts. Though starting in one-story shacks in the piney woods of the Panhandle, Cracker cooking in Florida has evolved with our tastes and times and is now just as home in high-rise apartments along the glistening waterways. When supplies were limited and the workday arduous, black coffee with leftover cornbread might serve as breakfast. Today’s bounty and life’s relative ease bring mornings with lattes and biscotti, biscuits and sausage gravy. What’s on the plate has changed, but our heritage infuses who we are. As we follow the path laid out by gastronomic pioneers, this culinary quest, guided by sixth-generation Cracker Joy Sheffield Harris, will whet your appetite with recipes and sumptuous reflections. Pull up a chair and dig in. |
| the tools of mathematical reasoning lakins solutions: Read, Write, Lead Regie Routman, 2014-06-17 Literacy is a skill for all time, for all people. It is an integral part of our lives, whether we are students or adult professionals. Giving all educators the breadth of knowledge and practical tools that help students strengthen their literacy skills is the focus of Read, Write, Lead. Drawing on her experience as a mentor teacher, reading specialist, instructional coach, and staff developer, author Regie Routman offers time-tested advice on how to develop a schoolwide learning culture that leads to more effective reading and writing across the curriculum. She explains how every school—including yours—can: implement instructional practices that lead to better engagement and achievement in reading and writing for all students, from kindergarten through high school, including second-language and struggling learners; build Professional Literacy Communities of educators working together to create sustainable school change through professional learning based on shared beliefs; reduce the need for intervention through daily practices that ensure success, even for our most vulnerable learners; and embed the language of productive feedback in responsive instruction, conferences, and observations in order to accelerate learning for students, teachers, and leaders. In their own voices, teachers, principals, literacy specialists, and students offer real-life examples of changes that led to dramatic improvement in literacy skills and—perhaps just as important--increased joy in teaching and learning. Scattered throughout the book are “Quick Wins”--ideas and actions that can yield positive, affirming results while tackling the tough work of long-term change. |
| the tools of mathematical reasoning lakins solutions: Baseball's Forgotten Heroes Tony Salin, 1999 Focusing on such athletes as Art Pennington, Bruno Haas, and Bill Lange, Salin presents the stories of more than a dozen former players, many in his own words. 15 photos. |
| the tools of mathematical reasoning lakins solutions: Complexity Carlos Gershenson, 2008 Contributions de : Peter M. Allen, Philip W. Anderson, W. Brian Arthur, Yaneer Bar-Yam, Eric Bonabeau, Paul Cilliers, Jim Crutchfeld, Bruce Edmonds, Nigel Gilbert, Hermann Haken, Francis Heylighen, Bernardo A. Huberman, Stuart A. Kaufman, Seth Lloyd, Gottfried Mayer-Kress, Melanie Mitchell, Edgar Morin, Mark Newman, Grégoire Nicolis, Jordan B. Pollack, Peter Schuster, Ricard V. Solé, Tamás Vicsek, Stephen Wolfram. |
| the tools of mathematical reasoning lakins solutions: How to Write Mathematics Norman Earl Steenrod, 1973 This book contains four essays on expository writing of books and papers at the research level and at the level of graduate texts. The authors were the four members of the AMS Committee on Expository Writing. |
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Review of best AI summarizers : r/NoteTaking - Reddit
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You can use Classroom in your school to streamline assignments, boost collaboration, and foster communication. Classroom is available on the web or by mobile app. You can use Classroom …
Review of best AI summarizers : r/NoteTaking - Reddit
May 28, 2024 · Has a good user-friendly interface, multiple language options, and integrates with other writing tools. Downside is it's priced as a B2B tool so it's quite expensive. QuillBot is …
any experience with Vevor? : r/Tools - Reddit
This sub is for tool enthusiasts worldwide to talk about tools, professionals and hobbyists alike. We welcome posts about "new tool day", estate sale/car boot sale finds, "what is this" tool, …
Please recommend a "free" data recovery software with the
Jun 7, 2022 · There are higher caliber tools for severe logical damage, but none of them are nearly as generous in their free trial. If the drive is failing or you’ve deleted data from a TRIM …
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