In this paper, we prove there exist at least n+12+1 geometrically distinct closed characteristics on every compact convex hypersurface Σ in R2n, where n≥2. In particular, this gives a new proof in ...the case n=3 to a long standing conjecture in Hamiltonian analysis. Moreover, there exist at least n2+1 geometrically distinct non-hyperbolic closed characteristics on Σ provided the number of geometrically distinct closed characteristics on Σ is finite.
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42.
Desart Fractal Goldyuk, Arthur
Journal of new frontiers in spatial concepts,
12/2018, Volume:
10
Journal Article
Open access
Developing “space structure” geometry for skyscraper, based on mixed uses of open spaces and closed volumes.
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We construct complex contact structures on C 2 n + 1 for any n ≥ 1 with the property that every holomorphic Legendrian map C → C 2 n + 1 is constant. In particular, these contact structures are not ...globally contactomorphic to the standard complex contact structure on C 2 n + 1 .
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The use of power law forms to describe hydraulic geometry is a classic subject with a history of over 70 years. Two distinct forms of power laws have been proposed: at‐a‐station hydraulic geometry ...(AHG) and downstream hydraulic geometry (DHG). Although the utility of these semiempirical expressions is widely recognized, they remain poorly understood in terms of the mechanisms underlying the differences between AHG and DHG, as well as the variability among different systems. In this study, we attempt to address these basic issues. Two hypotheses are proposed: (a) the different geomorphic relationships represented by AHG and DHG result from the control of lateral adjustment of the bank and flow turbulence over short and long timescales, respectively; and (b) the systematic variability of the AHG and DHG exponents is related to the description of the frictional resistance. These two hypotheses are embedded in our theoretical models and lead to explicit functional forms for AHG and DHG. The verification of our hypotheses is based on a large data set consisting of over 550 b‐f‐m exponents and 120 power law hydraulic relations. The analysis highlights the role of uncertainties in data acquisition and theoretical/statistical explanations. In addition, the theoretical expressions of AHG also provide an explanation of at‐many‐stations hydraulic geometry (AMHG) in a physical sense. Overall, our work provides new insights into the fundamental theory of power laws and hydraulic geometry.
Key Points
Theoretical expressions of both at‐a‐station hydraulic geometry (AHG) and downstream hydraulic geometry (DHG) are derived
The theoretical expressions of AHG provides a physical explanation for at‐many‐stations hydraulic geometry
The different behaviors of AHG and DHG are attributed to the lateral adjustment of the bank and flow turbulence, respectively
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Triangulations De Loera, Jesus
2010, 2010-08-16, Volume:
25
eBook
Triangulations presents the first comprehensive treatment of the theory of secondary polytopes and related topics. The text discusses the geometric structure behind the algorithms and shows new ...emerging applications, including hundreds of illustrations, examples, and exercises.
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We study a new class of square Sierpinski carpets formula omitted: see PDF ( formula omitted: see PDF ) on formula omitted: see PDF , which are not quasisymmetrically equivalent to the standard ...Sierpinski carpets. We prove that the group of quasisymmetric self-maps of each formula omitted: see PDF is the Euclidean isometry group of formula omitted: see PDF . We also establish that formula omitted: see PDF and formula omitted: see PDF are quasisymmetrically equivalent if and only if formula omitted: see PDF .
The aim of the present paper is the classification of real hypersurfaces M equipped with the condition Al = lA, l = R(.,...)..., restricted in a subspace of the tangent space T p M of M at a point p. ...This class is large and difficult to classify, therefore a second condition is imposed: (∇...l)X =...(X)...+...(X)lX, where...(X),...(X) are 1-forms. The last condition is studied for the first time and is much weaker than ∇...l = 0 which has been studied so far. The Jacobi Structure Operator satisfying this weaker condition can be called generalized...-parallel Jacobi Structure Operator.
In this paper, we introduce a new approach based on distance fields to exactly impose boundary conditions in physics-informed deep neural networks. The challenges in satisfying Dirichlet boundary ...conditions in meshfree and particle methods are well-known. This issue is also pertinent in the development of physics informed neural networks (PINN) for the solution of partial differential equations. We introduce geometry-aware trial functions in artificial neural networks to improve the training in deep learning for partial differential equations. To this end, we use concepts from constructive solid geometry (R-functions) and generalized barycentric coordinates (mean value potential fields) to construct ϕ(x), an approximate distance function to the boundary of a domain in Rd. To exactly impose homogeneous Dirichlet boundary conditions, the trial function is taken as ϕ(x) multiplied by the PINN approximation, and its generalization via transfinite interpolation is used to a priori satisfy inhomogeneous Dirichlet (essential), Neumann (natural), and Robin boundary conditions on complex geometries. In doing so, we eliminate modeling error associated with the satisfaction of boundary conditions in a collocation method and ensure that kinematic admissibility is met pointwise in a Ritz method. With this new ansatz, the training for the neural network is simplified: sole contribution to the loss function is from the residual error at interior collocation points where the governing equation is required to be satisfied. Numerical solutions are computed using strong form collocation and Ritz minimization. To convey the main ideas and to assess the accuracy of the approach, we present numerical solutions for linear and nonlinear boundary-value problems over convex and nonconvex polygonal domains as well as over domains with curved boundaries. Benchmark problems in one dimension for linear elasticity, advection-diffusion, and beam bending; and in two dimensions for the steady-state heat equation, Laplace equation, biharmonic equation (Kirchhoff plate bending), and the nonlinear Eikonal equation are considered. The construction of approximate distance functions using R-functions extends to higher dimensions, and we showcase its use by solving a Poisson problem with homogeneous Dirichlet boundary conditions over the four-dimensional hypercube. The proposed approach consistently outperforms a standard PINN-based collocation method, which underscores the importance of exactly (a priori) satisfying the boundary condition when constructing a loss function in PINN. This study provides a pathway for meshfree analysis to be conducted on the exact geometry without domain discretization.
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This volume contains contributions from speakers at the 2015-2018 joint Johns Hopkins University and University of Maryland Complex Geometry Seminar. It begins with a survey article on recent ...developments in pluripotential theory and its applications to Kähler-Einstein metrics and continues with articles devoted to various aspects of the theory of complex manifolds and functions on such manifolds.
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