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"Geometry, Solid."
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Get in shape : two-dimensional and three-dimensional shapes
\"Introduces the reader to two-dimensional and three-dimensional shapes.\"-- Provided by publisher.
Book
Solid Geometry with MATLAB Programming
Solid geometry is defined as the study of the geometry of three-dimensional solid figures in Euclidean space. There are numerous techniques in solid geometry, mainly analytic geometry and methods using vectors, since they use linear equations and matrix algebra. Solid geometry is quite useful in everyday life, for example, to design different signs and symbols such as octagon shape stop signs, to indicate traffic rules, to design different 3D objects like cubicles in gaming zones, innovative lifts, creative 3D interiors, and to design 3D computer graphics. Studying solid geometry helps students to improve visualization and increase logical thinking and creativity since it is applicable everywhere in day-to-day life. It builds up a foundation for advanced levels of mathematical studies. Numerous competitive exams include solid geometry since its foundation is required to study other branches like civil engineering, mechanical engineering, computer science engineering, architecture, etc. This book is designed especially for students of all levels, and can serve as a fundamental resource for advanced level studies not only in mathematics but also in various fields like engineering, interior design, architecture, etc. It includes theoretical aspects as well as numerous solved examples. The book includes numerical problems and problems of construction as well as practical problems as an application of the respective topic. A special feature of this book is that it includes solved examples using the mathematical tool MATLAB.
eBook
Mummy math : an adventure in geometry
Two children must call on their knowledge of geometry to get out of a pyramid in Egypt.
Book
Shifting Spheres
The aim is to offer new tools for anunderstanding of the transformation of thepublic sphere, to cross different concepts(or sspheres') of the public sphere for abetter understanding of it. As such, theissue also is a state-of-the-art presentationof
eBook
Round
\"Sidman invites readers to search their worlds for round objects in nature. Illustrated with ... art by ... Taeeun Yoo, this fresh celebration shows why we love this shape best\"-- Provided by publisher.
Book
Euclidean distance-optimized data transformation for cluster analysis in biomedical data (EDOtrans)
Background
Data transformations are commonly used in bioinformatics data processing in the context of data projection and clustering. The most used Euclidean metric is not scale invariant and therefore occasionally inappropriate for complex, e.g., multimodal distributed variables and may negatively affect the results of cluster analysis. Specifically, the squaring function in the definition of the Euclidean distance as the square root of the sum of squared differences between data points has the consequence that the value 1 implicitly defines a limit for distances within clusters versus distances between (inter-) clusters.
Methods
The Euclidean distances within a standard normal distribution (N(0,1)) follow a N(0,
2
) distribution. The EDO-transformation of a variable X is proposed as
E
D
O
=
X
/
(
2
·
s
)
following modeling of the standard deviation
s
by a mixture of Gaussians and selecting the dominant modes via item categorization. The method was compared in artificial and biomedical datasets with clustering of untransformed data, z-transformed data, and the recently proposed pooled variable scaling.
Results
A simulation study and applications to known real data examples showed that the proposed EDO scaling method is generally useful. The clustering results in terms of cluster accuracy, adjusted Rand index and Dunn’s index outperformed the classical alternatives. Finally, the EDO transformation was applied to cluster a high-dimensional genomic dataset consisting of gene expression data for multiple samples of breast cancer tissues, and the proposed approach gave better results than classical methods and was compared with pooled variable scaling.
Conclusions
For multivariate procedures of data analysis, it is proposed to use the EDO transformation as a better alternative to the established z-standardization, especially for nontrivially distributed data. The “EDOtrans” R package is available at
https://cran.r-project.org/package=EDOtrans
.
Journal Article
I see 3-D
\"Join a group of friends as they spot 3-D shapes around them, and learn how the shapes are used in different and exciting ways\"-- Provided by publisher.
Book
Quasi-linear perturbations of Hamiltonian Klein-Gordon equations on spheres
The Hamiltonian \\int_X(\\lvert{\\partial_t u}\\rvert^2 + \\lvert{\\nabla u}\\rvert^2 + \\mathbf{m}^2\\lvert{u}\\rvert^2)\\,dx, defined on functions on \\mathbb{R}\\times X, where X is a compact manifold, has critical points which are solutions of the linear Klein-Gordon equation. The author considers perturbations of this Hamiltonian, given by polynomial expressions depending on first order derivatives of u. The associated PDE is then a quasi-linear Klein-Gordon equation. The author shows that, when X is the sphere, and when the mass parameter \\mathbf{m} is outside an exceptional subset of zero measure, smooth Cauchy data of small size \\epsilon give rise to almost global solutions, i.e. solutions defined on a time interval of length c_N\\epsilon^{-N} for any N. Previous results were limited either to the semi-linear case (when the perturbation of the Hamiltonian depends only on u) or to the one dimensional problem. The proof is based on a quasi-linear version of the Birkhoff normal forms method, relying on convenient generalizations of para-differential calculus.
eBook
The Soma puzzle book : a new approach to the classic pieces
\"A new twist on a popular puzzle cube! Invented by Piet Hein, the Soma cube is one of the most famous mechanical puzzles in the world. The traditional challenge and outcome is to build a cube or other structures from all seven pieces. Change the rules, change the outcome! The puzzles in this unique Soma-inspired collection are no longer predicated upon using all seven pieces at one time! By varying the number of pieces, there are many new and versatile puzzle challenges, from all types of recreational mathematics fields\"-- Provided by publisher.
Book
First-Order Axiom Systems Esub.d and Esub.da Extending Tarski's Esub.2 with Distance and Angle Function Symbols for Quantitative Euclidean Geometry
Tarski’s first-order axiom system E[sub.2] for Euclidean geometry is notable for its completeness and decidability. However, the Pythagorean theorem—either in its modern algebraic form a[sup.2] +b[sup.2] =c[sup.2] or in Euclid’s Elements —cannot be directly expressed in E[sub.2] , since neither distance nor area is a primitive notion in the language of E[sub.2] . In this paper, we introduce an alternative axiom system E[sub.d] in a two-sorted language, which takes a two-place distance function d as the only geometric primitive. We also present a conservative extension E[sub.da] of it, which also incorporates a three-place angle function a , both formulated strictly within first-order logic. The system E[sub.d] has two distinctive features: it is simple (with a single geometric primitive) and it is quantitative. Numerical distance can be directly expressed in this language. The Axiom of Similarity plays a central role in E[sub.d] , effectively killing two birds with one stone: it provides a rigorous foundation for the theory of proportion and similarity, and it implies Euclid’s Parallel Postulate (EPP). The Axiom of Similarity can be viewed as a quantitative formulation of EPP. The Pythagorean theorem and other quantitative results from similarity theory can be directly expressed in the languages of E[sub.d] and E[sub.da] , motivating the name Quantitative Euclidean Geometry . The traditional analytic geometry can be united under synthetic geometry in E[sub.d] . Namely, analytic geometry is not treated as a model of E[sub.d] , but rather, its statements can be expressed as first-order formal sentences in the language of E[sub.d] . The system E[sub.d] is shown to be consistent, complete, and decidable. Finally, we extend the theories to hyperbolic geometry and Euclidean geometry in higher dimensions.
Journal Article