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Register a new userThe method and examples of solving problems analogous to the problem of three bisectors
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S. F. Osinkin
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We suggest a method of solving the problem of existence of a triangle with prescribed two bisectors and one third element which can be taken as one of the angles, the sides, the heights or the medians, or the third bisector.
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Contact algorithm between different bodies plays an important role in solving collision problems. Usually it is not easy to be treated very well. Several ones for material point method were proposed by Bardenhangen, Brackbill, and Sulskycite{Bardenhagen2000,Bardenhagen2001}, Hu and Chencite{Hu_Chen2003}. An improved one for three-dimensional material point method is presented in this paper. The improved algorithm emphasizes the energy conservation of the system and faithfully recovers opposite acting forces between contacting bodies. Contrasted to the one by Bardenhagen, both the normal and tangential contacting forces are more appropriately applied to the contacting bodies via the contacting nodes of the background mesh; Contrasted to the one by Hu and Chen, not only the tangential velocities but also the normal ones are handled separately in respective individual mesh. This treatment ensures not only the contact/sliding/separation procedure but also the friction between contacting bodies are recovered. The presented contact algorithm is validated via numerical experiments including rolling simulation, impact of elastic spheres, impact of a Taylor bar and impact of plastic spheres. The numerical results show that the multi-mesh material point method with the improved contact algorithm is more suitable for solving collision problems.
The unfolding program TRUEE is a software package for the numerical solution of inverse problems. The algorithm was first applied in the FORTRAN77 program RUN. RUN is an event-based unfolding algorithm which makes use of the Tikhonov regularization. It has been tested and compared to different unfolding applications and stood out with notably stable results and reliable error estimation. TRUEE is a conversion of RUN to C++, which works within the powerful ROOT framework. The program has been extended for more user-friendliness and delivers unfolding results which are identical to RUN. Beside the simplicity of the installation of the software and the generation of graphics, there are new functions, which facilitate the choice of unfolding parameters and observables for the user. In this paper, we introduce the new unfolding program and present its performance by applying it to two exemplary data sets from astroparticle physics, taken with the MAGIC telescopes and the IceCube neutrino detector, respectively.
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This is an English translation of G.N. Chebotarevs classical paper On the Problem of Resolvents, which was originally written in Russian and published in Vol. 114, No. 2 of the Scientific Proceedings of the V.I. Ulyanov-Lenin Kazan State University. In this paper, Chebotarev extends the method in Wimans On the Application of Tschirnhaus Transformations to the Reduction of Algebraic Equations to argue that the general polynomial of degree 21 admits a solution using algebraic functions of at most 15 variables. However, his and Wimans proofs assume that certain intersections in affine space are generic without proof.
Our paper offers an analysis of how Dante describes the tre giri (three rings) of the Holy Trinity in Paradiso 33 of the Divine Comedy. We point to the myriad possibilities Dante may have been envisioning when he describes his vision of God at this final stage in his journey. Saiber focuses on the features of shape, motion, size, color, and orientation that Dante details in describing the Trinity. Mbirika uses mathematical tools from topology (specifically, knot theory) and combinatorics to analyze all the possible configurations that have a specific layout of three intertwining circles which we find particularly compelling given Dantes description of the Trinity: the round figures arranged in a triangular format with rotational and reflective symmetry. Of the many possible link patterns, we isolate two particularly suggestive arrangements for the giri: the Brunnian link and the (3,3)-torus link. These two patterns lend themselves readily to a Trinitarian model.
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