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On the Brink of Paradox: Highlights from the Intersection of Philosophy and Mathematics (The\mit Press Ser.)
by Agustin RayoAn introduction to awe-inspiring ideas at the brink of paradox: infinities of different sizes, time travel, probability and measure theory, and computability theory.This book introduces the reader to awe-inspiring issues at the intersection of philosophy and mathematics. It explores ideas at the brink of paradox: infinities of different sizes, time travel, probability and measure theory, computability theory, the Grandfather Paradox, Newcomb's Problem, the Principle of Countable Additivity. The goal is to present some exceptionally beautiful ideas in enough detail to enable readers to understand the ideas themselves (rather than watered-down approximations), but without supplying so much detail that they abandon the effort. The philosophical content requires a mind attuned to subtlety; the most demanding of the mathematical ideas require familiarity with college-level mathematics or mathematical proof.The book covers Cantor's revolutionary thinking about infinity, which leads to the result that some infinities are bigger than others; time travel and free will, decision theory, probability, and the Banach-Tarski Theorem, which states that it is possible to decompose a ball into a finite number of pieces and reassemble the pieces so as to get two balls that are each the same size as the original. Its investigation of computability theory leads to a proof of Gödel's Incompleteness Theorem, which yields the amazing result that arithmetic is so complex that no computer could be programmed to output every arithmetical truth and no falsehood. Each chapter is followed by an appendix with answers to exercises. A list of recommended reading points readers to more advanced discussions. The book is based on a popular course (and MOOC) taught by the author at MIT.
On the Class Number of Abelian Number Fields: Extended with Tables by Ken-ichi Yoshino and Mikihito Hirabayashi
by Helmut HasseWith this translation, the classic monograph Über die Klassenzahl abelscher Zahlkörper by Helmut Hasse is now available in English for the first time.The book addresses three main topics: class number formulas for abelian number fields; expressions of the class number of real abelian number fields by the index of the subgroup generated by cyclotomic units; and the Hasse unit index of imaginary abelian number fields, the integrality of the relative class number formula, and the class number parity. Additionally, the book includes reprints of works by Ken-ichi Yoshino and Mikihito Hirabayashi, which extend the tables of Hasse unit indices and the relative class numbers to imaginary abelian number fields with conductor up to 100. The text provides systematic and practical methods for deriving class number formulas, determining the unit index and calculating the class number of abelian number fields. A wealth of illustrative examples, together with corrections and remarks on the original work, make this translation a valuable resource for today’s students of and researchers in number theory.
On the Edge of Infinity: Encounters with the Beauty of the Universe
by Stefan KleinFrom the award-winning, bestselling German science author Stefan Klein An original way into the most thought-provoking scientific theories and ideas, On The Edge of Infinity is the perfect read for those curious about the workings of the universe.*How can a hurricane can reveal the world's unpredictability?How can a greying beard might demonstrate the irreversibility of time?How do the exploits of burglars in New York and London demonstrate how everything can be in two places at once?Employing stories about simple everyday items or occurrences as analogies to illuminate counterintuitive realities behind the visible world, On The Edge of Infinity reveals the astonishing beauty of the universe. This book transforms a simple everyday thing such as a rose blossom, or a day of stormy weather, into a key to understanding the most complex ideas and theories in 21st century physics. Stefan Klein unpicks the complexities and intricacies of physics, from the answered questions to the dark corners of what we have yet to discover, making this an accessible read to those with no previous knowledge of the subject.
On the Edge: The Art of Risking Everything
by Nate SilverIn the bestselling The Signal and the Noise, Nate Silver showed how forecasting would define the age of Big Data. Now, in this timely and riveting new book, Silver investigates “the River,” the community of like-minded people whose mastery of risk allows them to shape—and dominate—so much of modern life. <P><P> These professional risk-takers—poker players and hedge fund managers, crypto true believers and blue-chip art collectors—can teach us much about navigating the uncertainty of the twenty-first century. By immersing himself in the worlds of Doyle Brunson, Peter Thiel, Sam Bankman-Fried, Sam Altman, and many others, Silver offers insight into a range of issues that affect us all, from the frontiers of finance to the future of AI. <P><P> Most of us don’t have traits commonly found in the River: high tolerance for risk, appreciation of uncertainty, affinity for numbers—paired with an instinctive distrust of conventional wisdom and a competitive drive so intense it can border on irrational. For those in the River, complexity is baked in, and the work is how to navigate it. People in the River have increasing amounts of wealth and power in our society, and understanding their mindset—and the flaws in their thinking— is key to understanding what drives technology and the global economy today. <P><P> Taking us behind the scenes from casinos to venture capital firms, and from the FTX inner sanctum to meetings of the effective altruism movement, On the Edge is a deeply reported, all-access journey into a hidden world of power brokers and risk-takers. <p> <b>New York Times Bestseller</b>
On the Emergence of Time and Space in Closed Quantum Systems (Springer Theses)
by Tommaso FavalliTime, space and entanglement are the main characters in this book. Their nature is still a great mystery in physics and we study here the possibility that these three phenomena are closely connected, showing how entanglement can be at the basis of the emergence of time and space within closed quantum systems. We revisit and extend the Page and Wootters theory that was originally introduced in order to describe the emergence of time through entanglement between subsystems in a globally static, quantum Universe. In the book, after providing a complete review of the salient aspects of the theory, we establish a connection with recent research on the foundations of statistical mechanics and we provide a new understanding of the thermalization process. Furthermore, we generalize the framework in order describe the spatial degree of freedom and we provide a model of 3+1 dimensional, quantum spacetime emerging from entanglement among different subsystems in a globally "timeless" and "positionless" Universe. Finally, via the Page and Wootters theory, the evolution of quantum clocks within a gravitational field is treated and a time dilation effect is obtained in agreement with the Schwarzschild solution.
On the Epistemology of Data Science: Conceptual Tools for a New Inductivism (Philosophical Studies Series #148)
by Wolfgang PietschThis book addresses controversies concerning the epistemological foundations of data science: Is it a genuine science? Or is data science merely some inferior practice that can at best contribute to the scientific enterprise, but cannot stand on its own? The author proposes a coherent conceptual framework with which these questions can be rigorously addressed. Readers will discover a defense of inductivism and consideration of the arguments against it: an epistemology of data science more or less by definition has to be inductivist, given that data science starts with the data. As an alternative to enumerative approaches, the author endorses Federica Russo’s recent call for a variational rationale in inductive methodology. Chapters then address some of the key concepts of an inductivist methodology including causation, probability and analogy, before outlining an inductivist framework. The inductivist framework is shown to be adequate and useful for an analysis of the epistemological foundations of data science. The author points out that many aspects of the variational rationale are present in algorithms commonly used in data science. Introductions to algorithms and brief case studies of successful data science such as machine translation are included. Data science is located with reference to several crucial distinctions regarding different kinds of scientific practices, including between exploratory and theory-driven experimentation, and between phenomenological and theoretical science. Computer scientists, philosophers and data scientists of various disciplines will find this philosophical perspective and conceptual framework of great interest, especially as a starting point for further in-depth analysis of algorithms used in data science.
On the Estimation of Multiple Random Integrals and U-Statistics
by Péter MajorThis work starts with the study of those limit theorems in probability theory for which classical methods do not work. In many cases some form of linearization can help to solve the problem, because the linearized version is simpler. But in order to apply such a method we have to show that the linearization causes a negligible error. The estimation of this error leads to some important large deviation type problems, and the main subject of this work is their investigation. We provide sharp estimates of the tail distribution of multiple integrals with respect to a normalized empirical measure and so-called degenerate U-statistics and also of the supremum of appropriate classes of such quantities. The proofs apply a number of useful techniques of modern probability that enable us to investigate the non-linear functionals of independent random variables. This lecture note yields insights into these methods, and may also be useful for those who only want some new tools to help them prove limit theorems when standard methods are not a viable option.
On the Fractal Language of Medicine (Fractional Order Thinking in Exploring the Frontiers of STEM)
by Bruce J. West W. Alan MutchOn the Fractal Language of Medicine bridges a very clear gap among the knowledge gained over the last 20 years in the physical and life sciences on network theory, organ synchronicity and communication, the understanding of fractal signatures in health and disease and the importance of fractional calculus in integrating these concepts.The authors opine that the field of medicine has not appreciated this hard-won knowledge and has suffered greatly as a result. This book addresses this perceived deficiency by introducing medical researchers, clinicians, residents, first-year medical students and members of allied fields to the work of the so-called hard sciences. It seeks to facilitate effective communication between empiricists and theorists by making interdisciplinary efforts to explain complex mathematical concepts to physicians and, equally important, to elucidate complex medical concepts to physicists or mathematicians.This book will be of great interest to medical students, professionals and academics, as well as students and researchers of applied mathematics, especially those interested in fractional calculus and fractals.
On the Geometry of Some Special Projective Varieties
by Francesco RussoProviding an introduction to both classical and modern techniques in projective algebraic geometry, this monograph treats the geometrical properties of varieties embedded in projective spaces, their secant and tangent lines, the behavior of tangent linear spaces, the algebro-geometric and topological obstructions to their embedding into smaller projective spaces, and the classification of extremal cases. It also provides a solution of Hartshorne's Conjecture on Complete Intersections for the class of quadratic manifolds and new short proofs of previously known results, using the modern tools of Mori Theory and of rationally connected manifolds. The new approach to some of the problems considered can be resumed in the principle that, instead of studying a special embedded manifold uniruled by lines, one passes to analyze the original geometrical property on the manifold of lines passing through a general point and contained in the manifold. Once this embedded manifold, usually of lower codimension, is classified, one tries to reconstruct the original manifold, following a principle appearing also in other areas of geometry such as projective differential geometry or complex geometry.
On the Higher-Order Sheffer Orthogonal Polynomial Sequences
by Daniel J. GaliffaOn the Higher-Order Sheffer Orthogonal Polynomial Sequences sheds light on the existence/non-existence of B-Type 1 orthogonal polynomials. This book presents a template for analyzing potential orthogonal polynomial sequences including additional higher-order Sheffer classes. This text not only shows that there are no OPS for the special case the B-Type 1 class, but that there are no orthogonal polynomial sequences for the general B-Type 1 class as well. Moreover, it is quite provocative how the seemingly subtle transition from the B-Type 0 class to the B-Type 1 class leads to a drastically more difficult characterization problem. Despite this issue, a procedure is established that yields a definite answer to our current characterization problem, which can also be extended to various other characterization problems as well. Accessible to undergraduate students in the mathematical sciences and related fields, This book functions as an important reference work regarding the Sheffer sequences. The author takes advantage of Mathematica 7 to display unique detailed code and increase the reader's understanding of the implementation of Mathematica 7 and facilitate further experimentation. In addition, this book provides an excellent example of how packages like Mathematica 7 can be used to derive rigorous mathematical results.
On the Hypotheses Which Lie at the Bases of Geometry
by Bernhard Riemannjürgen JostThis book presents William Clifford's English translation of BernhardRiemann's classic text together with detailed mathematical, historical andphilosophical commentary. The basic concepts and ideas, as well as theirmathematical background, are provided, putting Riemann's reasoning into themore general and systematic perspective achieved by later mathematicians andphysicists (including Helmholtz, Ricci, Weyl, and Einstein) on the basis of hisseminal ideas. Following a historical introduction that positions Riemann's workin the context of his times, the history of the concept of space in philosophy,physics and mathematics is systematically presented. A subsequent chapter onthe reception and influence of the text accompanies the reader from Riemann's timesto contemporary research. Notonly mathematicians and historians of the mathematical sciences, but alsoreaders from other disciplines or those with an interest in physics orphilosophy will find this work both appealing and insightful.
On the Learnability of Physically Unclonable Functions (T-Labs Series in Telecommunication Services)
by Fatemeh GanjiThis book addresses the issue of Machine Learning (ML) attacks on Integrated Circuits through Physical Unclonable Functions (PUFs). It provides the mathematical proofs of the vulnerability of various PUF families, including Arbiter, XOR Arbiter, ring-oscillator, and bistable ring PUFs, to ML attacks. To achieve this goal, it develops a generic framework for the assessment of these PUFs based on two main approaches. First, with regard to the inherent physical characteristics, it establishes fit-for-purpose mathematical representations of the PUFs mentioned above, which adequately reflect the physical behavior of these primitives. To this end, notions and formalizations that are already familiar to the ML theory world are reintroduced in order to give a better understanding of why, how, and to what extent ML attacks against PUFs can be feasible in practice. Second, the book explores polynomial time ML algorithms, which can learn the PUFs under the appropriate representation. More importantly, in contrast to previous ML approaches, the framework presented here ensures not only the accuracy of the model mimicking the behavior of the PUF, but also the delivery of such a model. Besides off-the-shelf ML algorithms, the book applies a set of algorithms hailing from the field of property testing, which can help to evaluate the security of PUFs. They serve as a “toolbox”, from which PUF designers and manufacturers can choose the indicators most relevant for their requirements. Last but not least, on the basis of learning theory concepts, the book explicitly states that the PUF families cannot be considered as an ultimate solution to the problem of insecure ICs. As such, it provides essential insights into both academic research on and the design and manufacturing of PUFs.
On the Logos: A Naïve View on Ordinary Reasoning and Fuzzy Logic (Studies in Fuzziness and Soft Computing #354)
by Enric TrillasThis book offers an inspiring and naïve view on language and reasoning. It presents a new approach to ordinary reasoning that follows the author’s former work on fuzzy logic. Starting from a pragmatic scientific view on meaning as a quantity, and the common sense reasoning from a primitive notion of inference, which is shared by both laypeople and experts, the book shows how this can evolve, through the addition of more and more suppositions, into various formal and specialized modes of precise, imprecise, and approximate reasoning. The logos are intended here as a synonym for rationality, which is usually shown by the processes of questioning, guessing, telling, and computing. Written in a discursive style and without too many technicalities, the book presents a number of reflections on the study of reasoning, together with a new perspective on fuzzy logic and Zadeh’s “computing with words” grounded in both language and reasoning. It also highlights some mathematical developments supporting this view. Lastly, it addresses a series of questions aimed at fostering new discussions and future research into this topic. All in all, this book represents an inspiring read for professors and researchers in computer science, and fuzzy logic in particular, as well as for psychologists, linguists and philosophers.
On the Mathematics of Modelling, Metamodelling, Ontologies and Modelling Languages
by Brian Henderson-SellersComputing as a discipline is maturing rapidly. However, with maturity often comes a plethora of subdisciplines, which, as time progresses, can become isolationist. The subdisciplines of modelling, metamodelling, ontologies and modelling languages within software engineering e.g. have, to some degree, evolved separately and without any underpinning formalisms. Introducing set theory as a consistent underlying formalism, Brian Henderson-Sellers shows how a coherent framework can be developed that clearly links these four, previously separate, areas of software engineering. In particular, he shows how the incorporation of a foundational ontology can be beneficial in resolving a number of controversial issues in conceptual modelling, especially with regard to the perceived differences between linguistic metamodelling and ontological metamodelling. An explicit consideration of domain-specific modelling languages is also included in his mathematical analysis of models, metamodels, ontologies and modelling languages. This encompassing and detailed presentation of the state-of-the-art in modelling approaches mainly aims at researchers in academia and industry. They will find the principled discussion of the various subdisciplines extremely useful, and they may exploit the unifying approach as a starting point for future research.
On the Measurement of Social Phenomena: A Methodological Approach (SpringerBriefs in Political Science)
by Marco DelmastroThis book explores the analysis of social phenomena, using a multidisciplinary approach while addressing statistical, economic, sociological, as well as psychological issues.The author presents a detailed account of the procedures and techniques used to gather, process, and analyze data. Topics covered include, but are not limited to survey data, content analysis data, data visualization, as well as data about crimes.The book addresses this methodological framework that drives applied social sciences in an applicative and simple way, by analyzing key social phenomena such as the threats to journalism, the so-called chilling effect, and the market for news. Finally, the author examines the data and measures of the recent COVID-19 pandemic.This book is a must-read for everybody interested in a better understanding of the methodological analysis of social phenomena, social and political methodology, and applied science in general.
On the Move to Meaningful Internet Systems. OTM 2017 Conferences: Confederated International Conferences: CoopIS, C&TC, and ODBASE 2017, Rhodes, Greece, October 23-27, 2017, Proceedings, Part II (Lecture Notes in Computer Science #10574)
by Hervé Panetto, Christophe Debruyne, Walid Gaaloul, Mike Papazoglou, Adrian Paschke, Claudio Agostino Ardagna and Robert MeersmanThis double volumes LNCS 10573-10574 constitutes the refereed proceedings of the Confederated International Conferences: Cooperative Information Systems, CoopIS 2017, Ontologies, Databases, and Applications of Semantics, ODBASE 2017, and Cloud and Trusted Computing, C&TC, held as part of OTM 2017 in October 2017 in Rhodes, Greece.The 61 full papers presented together with 19 short papers were carefully reviewed and selected from 180 submissions. The OTM program every year covers data and Web semantics, distributed objects, Web services, databases, information systems, enterprise workflow and collaboration, ubiquity, interoperability, mobility, grid and high-performance computing.
On the Move to Meaningful Internet Systems. OTM 2018 Conferences: Confederated International Conferences: CoopIS, C&TC, and ODBASE 2018, Valletta, Malta, October 22-26, 2018, Proceedings, Part II (Lecture Notes in Computer Science #11230)
by Dumitru Roman Christophe Debruyne Hervé Panetto Robert Meersman Claudio Agostino Ardagna Henderik A. ProperThis double volumes LNCS 11229-11230 constitutes the refereed proceedings of the Confederated International Conferences: Cooperative Information Systems, CoopIS 2018, Ontologies, Databases, and Applications of Semantics, ODBASE 2018, and Cloud and Trusted Computing, C&TC, held as part of OTM 2018 in October 2018 in Valletta, Malta. The 64 full papers presented together with 22 short papers were carefully reviewed and selected from 173 submissions. The OTM program every year covers data and Web semantics, distributed objects, Web services, databases, informationsystems, enterprise workflow and collaboration, ubiquity, interoperability, mobility, grid and high-performance computing.
On the Move to Meaningful Internet Systems: Confederated International Conferences: CoopIS, ODBASE, C&TC 2019, Rhodes, Greece, October 21–25, 2019, Proceedings (Lecture Notes in Computer Science #11877)
by Christophe Debruyne Hervé Panetto Robert Meersman Claudio Agostino Ardagna Dave Lewis Martin HeppThis volume LNCS 11877 constitutes the refereed proceedings of the Confederated International Conferences: Cooperative Information Systems, CoopIS 2019, Ontologies, Databases, and Applications of Semantics, ODBASE 2019, and Cloud and Trusted Computing, C&TC, held as part of OTM 2019 in October 2019 in Rhodes, Greece. The 38 full papers presented together with 8 short papers were carefully reviewed and selected from 156 submissions. The OTM program every year covers data and Web semantics, distributed objects, Web services, databases, informationsystems, enterprise workflow and collaboration, ubiquity, interoperability, mobility, grid and high-performance computing.
On the Move: Changing Mechanisms of Mexico-U.S. Migration
by Filiz GaripWhy do Mexicans migrate to the United States? Is there a typical Mexican migrant? Beginning in the 1970s, survey data indicated that the average migrant was a young, unmarried man who was poor, undereducated, and in search of better employment opportunities. This is the general view that most Americans still hold of immigrants from Mexico. On the Move argues that not only does this view of Mexican migrants reinforce the stereotype of their undesirability, but it also fails to capture the true diversity of migrants from Mexico and their evolving migration patterns over time.Using survey data from over 145,000 Mexicans and in-depth interviews with nearly 140 Mexicans, Filiz Garip reveals a more accurate picture of Mexico-U.S migration. In the last fifty years there have been four primary waves: a male-dominated migration from rural areas in the 1960s and '70s, a second migration of young men from socioeconomically more well-off families during the 1980s, a migration of women joining spouses already in the United States in the late 1980s and '90s, and a generation of more educated, urban migrants in the late 1990s and early 2000s. For each of these four stages, Garip examines the changing variety of reasons for why people migrate and migrants' perceptions of their opportunities in Mexico and the United States. Looking at Mexico-U.S. migration during the last half century, On the Move uncovers the vast mechanisms underlying the flow of people moving between nations.
On the Nature of Charge Density Waves, Superconductivity and Their Interplay in 1T-TiSe₂ (Springer Theses)
by Chuan ChenThis thesis presents analytical theoretical studies on the interplay between charge density waves (CDW) and superconductivity (SC) in the actively studied transition-metal dichalcogenide 1T-TiSe2. It begins by reapproaching a years-long debate over the nature of the phase transition to the commensurate CDW (CCDW) state and the role played by the intrinsic tendency towards excitonic condensation in this system. A Ginzburg-Landau phenomenological theory was subsequently developed to understand the experimentally observed transition from commensurate to incommensurate CDW (ICDW) order with doping or pressure, and the emergence of a superconducting dome that coexists with ICDW. Finally, to characterize microscopically the effects of the interplay between CDW and SC, the spectrum of CDW fluctuations beyond mean-field was studied in detail. In the aggregate, the work reported here provides an encompassing understanding of what are possibly key microscopic underpinnings of the CDW and SC physics in TiSe2.
On the Study and Difficulties of Mathematics
by Augustus De MorganOne of the twentieth century's most eminent mathematical writers, Augustus De Morgan enriched his expositions with insights from history and psychology. On the Study and Difficulties of Mathematics represents some of his best work, containing points usually overlooked by elementary treatises, and written in a fresh and natural tone that provides a refreshing contrast to the mechanical character of common textbooks.Presuming only a knowledge of the rules of algebra and Euclidean theorems, De Morgan begins with some introductory remarks on the nature and objects of mathematics. He discusses the concept of arithmetical notion and its elementary rules, including arithmetical reactions and decimal fractions. Moving on to algebra, he reviews the elementary principles, examines equations of the first and second degree, and surveys roots and logarithms. De Morgan's book concludes with an exploration of geometrical reasoning that encompasses the formulation and use of axioms, the role of proportion, and the application of algebra to the measurement of lines, angles, the proportion of figures, and surfaces.
On the Theory of Maass Wave Forms (Universitext)
by Tobias Mühlenbruch Wissam RajiThis textbook provides a rigorous analytical treatment of the theory of Maass wave forms. Readers will find this unified presentation invaluable, as it treats Maass wave forms as the central area of interest. Subjects at the cutting edge of research are explored in depth, such as Maass wave forms of real weight and the cohomology attached to Maass wave forms and transfer operators. Because Maass wave forms are given a deep exploration, this book offers an indispensable resource for those entering the field.Early chapters present a brief introduction to the theory of classical modular forms, with an emphasis on objects and results necessary to fully understand later material. Chapters 4 and 5 contain the book’s main focus: L-functions and period functions associated with families of Maass wave forms. Other topics include Maass wave forms of real weight, Maass cusp forms, and weak harmonic Maass wave forms. Engaging exercises appear throughout the book, with solutions available online.On the Theory of Maass Wave Forms is ideal for graduate students and researchers entering the area. Readers in mathematical physics and other related disciplines will find this a useful reference as well. Knowledge of complex analysis, real analysis, and abstract algebra is required.
On-Orbit Operations Optimization
by Leping Yang Yanwei Zhu Xianhai Ren Yuanwen ZhangOn-orbit operations optimization among multiple cooperative or noncooperative spacecraft, which is often challenged by tight constraints and shifting parameters, has grown to be a hot issue in recent years. The authors of this book summarize related optimization problems into four planning categories: spacecraft multi-mission planning, far-range orbital maneuver planning, proximity relative motion planning and multi-spacecraft coordinated planning. The authors then formulate models, introduce optimization methods, and investigate simulation cases that address problems in these four categories. This text will serve as a quick reference for engineers, graduate students, postgraduates in the fields of optimization research and on-orbit operation mission planning.
On-the-Job Math Mysteries: Real-Life Math From Exciting Careers (Grades 4-8)
by Marya Washington TylerThe author of the best-selling Real Life Math Mysteries offers students more close encounters with engaging professionals! After working through the math problems encountered on the job, your students will have no doubt about the absolute importance of math in the real world. Just like a field trip, but a lot more practical, this book takes students out into the field where they count fish, direct air traffic, carve a totem pole, write software code, care for orphaned sea creatures, recover sunken valuables, mush dogs, salvage timber, feed the homeless, film wildlife, and a lot more.The math problems are tied to NCTM standards, and students will use skills such as selecting an operation, determining place value, using fractions and decimals, working with geometry, applying measurement skills, estimating, and recording and analyzing data to solve them. Each set of problems is presented on separate reproducible worksheets, and the book includes a list of skills and comprehensive answer key.Working with the engaging and diverse mathematics presented here, students will see exactly what is at stake in problem solving, why mathematics is essential, and why accuracy matters. The light will go on for your students as they see that mathematical skills are essential to following their dreams.Grades 4-8
OnRamp to Algebra: Extra Practice Workbook (Grade 7-9)
by Prentice HallOnramp To Algebra 2013 Extra Practice Workbook Grade 7/9