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"Differential calculus."
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On the differential structure of metric measure spaces and applications
The main goals of this paper are: (i) To develop an abstract differential calculus on metric measure spaces by investigating the
duality relations between differentials and gradients of Sobolev functions. This will be achieved without calling into play any sort of
analysis in charts, our assumptions being: the metric space is complete and separable and the measure is Radon and non-negative. (ii) To
employ these notions of calculus to provide, via integration by parts, a general definition of distributional Laplacian, thus giving a
meaning to an expression like
eBook
Derivative with a New Parameter
Derivative with a New Parameter: Theory, Methods and Applications discusses the first application of the local derivative that was done by Newton for general physics, and later for other areas of the sciences.The book starts off by giving a history of derivatives, from Newton to Caputo.
eBook
Boolean differential calculus
The Boolean Differential Calculus (BDC) is a very powerful theory that extends the basic concepts of Boolean Algebras significantly. Its applications are based on Boolean spaces B and B n , Boolean operations, and basic structures such as Boolean Algebras and Boolean Rings, Boolean functions, Boolean equations, Boolean inequalities, incompletely specified Boolean functions, and Boolean lattices of Boolean functions. These basics, sometimes also called switching theory, are widely used in many modern information processing applications. The BDC extends the known concepts and allows the consideration of changes of function values. Such changes can be explored for pairs of function values as well as for whole subspaces. The BDC defines a small number of derivative and differential operations. Many existing theorems are very welcome and allow new insights due to possible transformations of problems. The available operations of the BDC have been efficiently implemented in several software packages. The common use of the basic concepts and the BDC opens a very wide field of applications. The roots of the BDC go back to the practical problem of testing digital circuits. The BDC deals with changes of signals which are very important in applications of the analysis and the synthesis of digital circuits. The comprehensive evaluation and utilization of properties of Boolean functions allow, for instance, to decompose Boolean functions very efficiently; this can be applied not only in circuit design, but also in data mining. Other examples for the use of the BDC are the detection of hazards or cryptography. The knowledge of the BDC gives the scientists and engineers an extended insight into Boolean problems leading to new applications, e.g., the use of Boolean lattices of Boolean functions.
Book
Automorphisms of Two-Generator Free Groups and Spaces of Isometric Actions on the Hyperbolic Plane
The automorphisms of a two-generator free group \\mathsf F_2 acting on the space of orientation-preserving isometric actions of \\mathsf F_2 on hyperbolic 3-space defines a dynamical system. Those actions which preserve a hyperbolic plane but not an orientation on that plane is an invariant subsystem, which reduces to an action of a group \\Gamma on \\mathbb R ^3 by polynomial automorphisms preserving the cubic polynomial \\kappa _\\Phi (x,y,z) := -x^{2} -y^{2} + z^{2} + x y z -2 and an area form on the level surfaces \\kappa _{\\Phi}^{-1}(k).
eBook
Kirchhoff–Love shell theory based on tangential differential calculus
The Kirchhoff–Love shell theory is recasted in the frame of the tangential differential calculus (TDC) where differential operators on surfaces are formulated based on global, three-dimensional coordinates. As a consequence, there is no need for a parametrization of the shell geometry implying curvilinear surface coordinates as used in the classical shell theory. Therefore, the proposed TDC-based formulation also applies to shell geometries which are zero-isosurfaces as in the level-set method where no parametrization is available in general. For the discretization, the TDC-based formulation may be used based on surface meshes implying element-wise parametrizations. Then, the results are equivalent to those obtained based on the classical theory. However, it may also be used in recent finite element approaches as the TraceFEM and CutFEM where shape functions are generated on a background mesh without any need for a parametrization. Numerical results presented herein are achieved with isogeometric analysis for classical and new benchmark tests. Higher-order convergence rates in the residual errors are achieved when the physical fields are sufficiently smooth.
Journal Article
Banach manifold structure and infinite-dimensional analysis for causal fermion systems
A mathematical framework is developed for the analysis of causal fermion systems in the infinite-dimensional setting. It is shown that the regular spacetime point operators form a Banach manifold endowed with a canonical Fréchet-smooth Riemannian metric. The so-called expedient differential calculus is introduced with the purpose of treating derivatives of functions on Banach spaces which are differentiable only in certain directions. A chain rule is proven for Hölder continuous functions which are differentiable on expedient subspaces. These results are made applicable to causal fermion systems by proving that the causal Lagrangian is Hölder continuous. Moreover, Hölder continuity is analyzed for the integrated causal Lagrangian.
Journal Article