Using the coherent state techniques developed for the analysis of the EPRL model we give the asymptotic formula for the Ponzano–Regge model amplitude for non-tardis triangulations of handlebodies in the limit of large boundary spins. The formula produces a sum over all possible immersions of the boundary triangulation and its value is given by the cosine of the Regge action evaluated on these. Furthermore the asymptotic scaling registers the existence of flexible immersions. We verify numerically that this formula approximates the 6j-symbol for large spins.

https://iopscience.iop.org/article/10.1088/1751-8113/43/11/115203/pdf

Asymptotic analysis of the Ponzano-Regge model for handlebodies

A NEW textbook supplying a real need is always welcome. The book now under review is based on lectures given in the University of Cambridge and is essentially designed for the use of students, but will be found to be of great service to astronomers also. Modern developments in astronomy have rendered the older textbooks incomplete, so in addition to the discussion of the usual problems of spherical astronomy, the book contains much new matter, including the essential discussion of such subjects as heliocentric co-ordinates, proper motions, the orbits of binary stars, and the use of photography in precise astronomical measurements, information on which is only to be found in scattered scientific papers.Text-Book on Spherical Astronomy.By Prof. W. M. Smart. Pp. xi + 414. (Cambridge: At the University Press, 1931.) 21s. net.

Text Book On Spherical Astronomy

Is there anything interesting left in isometric embeddings after the problem had been solved by John Nash? We do not venture a definite answer, but we outline the boundary of our knowledge and indicate conjectural directions one may pursue further. Our presentation is by no means comprehensive. The terrain of isometric embeddings and the fields surrounding this terrain are vast and craggy with valleys separated by ridges of unreachable mountains; people cultivating their personal gardens in these “valleys” only vaguely aware of what happens away from their domains and the authors of general accounts on isometric embeddings have a limited acquaintance with the original papers. Even the highly cited articles by Nash have been carefully read only by a handful of

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Geometric, algebraic, and analytic descendants of Nash isometric embedding theorems

We propose a new treatment of the quantum simplicity constraints appearing in the general ${SO(D+1)}$ formulation of loop quantum gravity for the ${(1+D)}$-dimensional space-time. Instead of strongly imposing the constraints, we construct a specific form of weak solutions by employing the spin net-work states with specific ${SO(D+1)}$ coherent intertwiners. These states weakly satisfy the quantum simplicity constraint via the vanishing expectation values, and the quantum Gaussian constraints can be imposed strongly. Remarkably, those specific ${SO(D+1)}$ coherent intertwiners used to construct our solutions have natural interpretations of the $D$-dimensional polytopes, commonly viewed as basic units of the discrete spatial geometry. Therefore, while the strong imposition of the quantum simplicity constraints leads to an over-constrained solution space, our weak solution space for the constraints may contain the correct semiclassical degrees of freedom for intrinsic geometry of the spatial hypersurfaces. Moreover, some concrete relations are established between our construction and other existing approaches in solving the simplicity constraints in all dimensional loop quantum gravity, providing valuable insights into this unresolved important issue.

Coherent intertwiner solution of simplicity constraint in all dimensional loop quantum gravity

By analogy with the solution of triangles, the solution of polyhedra means a theory and methods for calculating some geometric parameters of polyhedra in terms of other parameters of them. The main content of this paper is a survey of results on calculating the volumes of polyhedra in terms of their metrics and combinatorial structures. It turns out that a far-reaching generalization of Heron's formula for the area of a triangle to the volumes of polyhedra is possible, and it underlies the proof of the conjecture that the volume of a deformed flexible polyhedron remains constant. Bibliography: 110 titles.

https://iopscience.iop.org/article/10.1070/RM2011v066n03ABEH004748/pdf

Algebraic methods for solution of polyhedra

We prove that the Dehn invariants of any Bricard octahedron remain constant during the flex and that the Strong Bellows Conjecture holds true for the Steffen flexible polyhedron.

The Dehn invariants of the Bricard octahedra

Using Green's theorem we reduce the variation of the total mean curvature of a smooth surface in the Euclidean 3-space to a line integral of a special vector field and obtain the following well-known theorem as an immediate consequence: the total mean curvature of a closed smooth surface in the Euclidean 3-space is stationary under an infinitesimal flex.

On the total mean curvature of non-rigid surfaces

Two basic theorems of the theory of flexible polyhedra were proven by completely different methods: Alexander used analysis, namely, the Stokes theorem, to prove that the total mean curvature remains constant during the flex, while Sabitov used algebra, namely, the theory of resultants, to prove that the oriented volume remains constant during the flex. We show that none of these methods can be used to prove both theorems. As a by-product, we prove that the total mean curvature of any polyhedron in the Euclidean 3-space is not an algebraic function of its edge lengths.

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Algebra versus analysis in the theory of flexible polyhedra

Classical H. Minkowski theorems on existence and uniqueness of convex polyhedra with prescribed directions and areas of faces as well as the well-known generalization of H. Minkowski uniqueness theorem due to A.D. Alexandrov are extended to a class of nonconvex polyhedra which are called polyhedral herissons and may be described as polyhedra with injective spherical image.

/pdf/minkowski-type-and-alexandrov-type-theorems-for-polyhedral-1p9opbce78.pdf

Minkowski-type and Alexandrov-Type Theorems for Polyhedral Herissons

Two basic theorems of the theory of flexible polyhedra were proven by completely different methods: R. Alexander used analysis, namely, the Stokes theorem, to prove that the total mean curvature remains constant during the flex, while I.Kh. Sabitov used algebra, namely, the theory of resultants, to prove that the oriented volume remains constant during the flex. We show that none of these methods can be used to prove the both theorems. As a by-product, we prove that the total mean curvature of any polyhedron in the Euclidean 3-space is not an algebraic function of its edge lengths.

Victor Alexandrov

Papers

The Dehn invariants of the Bricard octahedra

On the total mean curvature of non-rigid surfaces

Algebra versus analysis in the theory of flexible polyhedra

Minkowski-type and Alexandrov-Type Theorems for Polyhedral Herissons

Algebra versus analysis in the theory of flexible polyhedra