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Register a new userGeometry, Analysis and Morphogenesis: Problems and Prospects
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The remarkable range of biological forms in and around us, such as the undulating shape of a leaf or flower in the garden, the coils in our gut, or the folds in our brain, raise a number of questions at the interface of biology, physics and mathematics. How might these shapes be predicted, and how can they eventually be designed? We review our current understanding of this problem, that brings together analysis, geometry and mechanics in the description of the morphogenesis of low-dimensional objects. Starting from the view that shape is the consequence of metric frustration in an ambient space, we examine the links between the classical Nash embedding problem and biological morphogenesis. Then, motivated by a range of experimental observations and numerical computations, we revisit known rigorous results on curvature-driven patterning of thin elastic films, especially the asymptotic behaviors of the solutions as the (scaled) thickness becomes vanishingly small and the local curvature can become large. Along the way, we discus open problems that include those in mathematical modeling and analysis along with questions driven by the allure of being able to tame soft surfaces for applications in science and engineering.
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The fundamental theorem of affine geometry is a classical and useful result. For finite-dimensional real vector spaces, the theorem roughly states that a bijective self-mapping which maps lines to lines is affine. In this note we prove several generalizations of this result and of its classical projective counterpart. We show that under a significant geometric relaxation of the hypotheses, namely that only lines parallel to one of a fixed set of finitely many directions are mapped to lines, an injective mapping of the space must be of a very restricted polynomial form. We also prove that under mild additional conditions the mapping is forced to be affine-additive or affine-linear. For example, we show that five directions in three dimensional real space suffice to conclude affine-additivity. In the projective setting, we show that n+2 fixed projective points in real n-dimensional projective space , through which all projective lines that pass are mapped to projective lines, suffice to conclude projective-linearity.
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Sergei A. Levshakov (National Astronomical Observatory
, Japan andn Ioffe Institute
, St. Petersburg
1997
The current status of extragalactic deuterium abundance is discussed using two examples of `low and `high D/H measurements. We show that the discordance of these two types of D abundances may be a consequence of the spatial correlations in the stochastic velocity field. Within the framework of the generalized procedure (accounting for such effects) one finds good agreement between different observations and the theoretical predictions for standard big bang nucleosynthesis (SBBN). In particular, we show that the deuterium absorption seen at z = 2.504 toward Q1009+2956 and the H+D Ly-alpha profile observed at z = 0.701 toward Q1718+4807 are compatible with D/H $sim 4.1 - 4.6times10^{-5}$. This result supports SBBN and, thus, no inhomogeneity is needed. The problem of precise D/H measurements is discussed.
We discuss 2f production at LC energies (f eq t). This type of reaction has a big event number and may give interesting hints to the existence and perhaps to details of New Physics like susy, LQ, Z, etc. {For} any search the radiative corrections have to be controlled carefully. An interesting challenge for theoreticians could also be the Giga-Z option of the TESLA project.
We prove well-posedness and regularity results for elliptic boundary value problems on certain domains with a smooth set of singular points. Our class of domains contains the class of domains with isolated oscillating conical singularities, and hence they generalize the classical results of Kondratiev on domains with conical singularities. The proofs are based on conformal changes of metric, on the differential geometry of manifolds with boundary and bounded geometry, and on our earlier results on manifolds with boundary and bounded geometry.
We show how one can associate to a given class of finite type G-structures a classifying Lie algebroid. The corresponding Lie groupoid gives models for the different geometries that one can find in the class, and encodes also the different types of symmetry groups.
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