An atlas is a collection of consistent coordinate charts on a manifold, where "consistent" most commonly means that the transition functions of the charts are smooth. As the name suggests, an atlas corresponds to a collection of maps, each of which shows a piece of a manifold and looks like flat Euclidean space. To use an atlas, one needs to know how the maps overlap. To be useful, the maps must not be too different on these overlapping areas.
A smooth atlas has transition functions that are C-infty smooth (i.e., infinitely differentiable). The consequence is that a smooth function on one chart is smooth in any other chart (by the chain rule for higher derivatives). Similarly, one could have an atlas in class , where the transition functions are in class C-k.
Atlas 2 For Mathematica
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In the even-dimensional case, one may ask whether the transition functions are holomorphic. In this case, one has a holomorphic atlas, and by the chain rule, it makes sense to ask if a function on the manifold is holomorphic.
It is possible for two atlases to be compatible, meaning the union is also an atlas. By Zorn's lemma, there always exists a maximal atlas, where a maximal atlas is an atlas not contained in any other atlas. However, in typical applications, it is not necessary to use a maximal atlas and any sufficiently refined atlas will do.
The tool gives access to Differential Geometry Library directly from Mathematica.The library has over 550 objects for differential geometry and its applications and frequently updated. The are hundreds of Exact Solutions of Einstein's Field Equations and atlas's graphical user interface (see below) allows calculate any of the objects/Exact Solutions just in few seconds.
GRTensorII[15] is a computer algebra package for performing calculations in the general area of differential geometry. Atlas 2 for Maple[16] is a modern differential geometry for Maple. DifferentialGeometry[17] is a package which performs fundamental operations of calculus on manifolds, differential geometry, tensor calculus, General Relativity, Lie algebras, Lie groups, transformation groups, jet spaces, and the variational calculus. It is included with Maple. Physics[18] is a package developed as part of Maple, which implements symbolic computations with most of the objects used in mathematical physics. It includes objects from general relativity (tensors, metrics, covariant derivatives, tetrads etc.), quantum mechanics (Kets, Bras, commutators, noncomutative variables) etc.
I am looking for a Mathematica package that can manipulate tensors for supergravity, string theory or M-theory. I am particularly looking for a package that can do spinor and Clifford algebra computations. Also, I would like this package to be able to do wedge and hodge dual, and other computation relating to forms.Can anyone suggest a specific one? I looked for atlas2, but it seems I have to pay to use it without a trial version.
Impaired function of masticatory muscles will lead to trismus. Routine delineation of these muscles during planning may improve dose tracking and facilitate dose reduction resulting in decreased radiation-related trismus. This study aimed to compare a deep learning model with a commercial atlas-based model for fast auto-segmentation of the masticatory muscles on head and neck computed tomography (CT) images.
The development of computational tools to automatically generate OAR contours can reduce the time and effort required for HNC contouring and plan development, as well as inter-observer contour variations. Specifically, organ auto-segmentation has been extensively studied [7,8,9,10] using both CT and MR image datasets [11, 12]. One approach, atlas-based auto-segmentation (ABAS) [13, 14], is a traditional method for organ contouring and various factors can affect segmentation performance. These include the size of the dataset used to create the atlas, approaches for image registration, and approaches for label fusion. Because the atlas size is fixed, the main limitation for ABAS is the ability to overcome variations in patient anatomy. In recent years, deep learning-based methods [15, 16] have shown great success for biomedical image segmentation and have been introduced to the field of head and neck anatomy segmentation. However, the literature is limited in assessing masticatory muscles (MMs) auto-segmentation [17, 18], which may be due to the lack of delineation guidelines for MMs.
The thefts took place between 1992 and 2017 and were unsophisticated: generally Priore would simply walk out of the R. Oliver Special Collections Room with the stolen materials.The thefts include George Washington's journal, a $1m copy of Newton's Principia Mathematica, and a $1.2m 19th century German explorer's atlas.
The Gurobi Optimizer is a mathematical optimization solver for a wide variety of problem types, including: linear programming (LP), quadratic programming (QP), quadratically constrained programming (QCP), mixed integer linear programming (MILP), mixed-integer quadratic programming (MIQP), and mixed-integer quadratically constrained programming (MIQCP).
Julia is a high-level, high-performance dynamic programming language for numerical computing. It provides a sophisticated compiler, distributed parallel execution, numerical accuracy, and an extensive mathematical function library. 2ff7e9595c
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