TY - JOUR A1 - Junek, Heinz T1 - Zyklizität in Raum, zeit und geist : über Pflasterungen, Rollkurven, Dezimalbrüche, Schwingungen, Wellen, Iteration und Neuronale Netze JF - Zyklizität & Rhythmik: eine multidisziplinäre Vorlesungsreihe Y1 - 2020 SN - 978-3-86464-169-5 SP - 85 EP - 103 PB - trafo CY - Berlin ER - TY - JOUR A1 - Denecke, Klaus-Dieter T1 - Partial clones JF - Asian-European journal of mathematics : AEJM N2 - A set C of operations defined on a nonempty set A is said to be a clone if C is closed under composition of operations and contains all projection mappings. The concept of a clone belongs to the algebraic main concepts and has important applications in Computer Science. A clone can also be regarded as a many-sorted algebra where the sorts are the n-ary operations defined on set A for all natural numbers n >= 1 and the operations are the so-called superposition operations S-m(n) for natural numbers m, n >= 1 and the projection operations as nullary operations. Clones generalize monoids of transformations defined on set A and satisfy three clone axioms. The most important axiom is the superassociative law, a generalization of the associative law. If the superposition operations are partial, i.e. not everywhere defined, instead of the many-sorted clone algebra, one obtains partial many-sorted algebras, the partial clones. Linear terms, linear tree languages or linear formulas form partial clones. In this paper, we give a survey on partial clones and their properties. KW - Operation KW - term KW - formula KW - superposition of operations KW - terms and KW - formulas KW - linear term KW - linear formula KW - linear tree language KW - clone KW - partial clone KW - linear hypersubstitution KW - dht-symmetric category KW - partial KW - theory Y1 - 2020 U6 - https://doi.org/10.1142/S1793557120501612 SN - 1793-5571 SN - 1793-7183 VL - 13 IS - 8 PB - World Scientific CY - Singapore ER - TY - JOUR A1 - Somogyvári, Márk A1 - Reich, Sebastian T1 - Convergence tests for transdimensional Markov chains in geoscience imaging JF - Mathematical geosciences : the official journal of the International Association for Mathematical Geosciences N2 - Classic inversion methods adjust a model with a predefined number of parameters to the observed data. With transdimensional inversion algorithms such as the reversible-jump Markov chain Monte Carlo (rjMCMC), it is possible to vary this number during the inversion and to interpret the observations in a more flexible way. Geoscience imaging applications use this behaviour to automatically adjust model resolution to the inhomogeneities of the investigated system, while keeping the model parameters on an optimal level. The rjMCMC algorithm produces an ensemble as result, a set of model realizations, which together represent the posterior probability distribution of the investigated problem. The realizations are evolved via sequential updates from a randomly chosen initial solution and converge toward the target posterior distribution of the inverse problem. Up to a point in the chain, the realizations may be strongly biased by the initial model, and must be discarded from the final ensemble. With convergence assessment techniques, this point in the chain can be identified. Transdimensional MCMC methods produce ensembles that are not suitable for classic convergence assessment techniques because of the changes in parameter numbers. To overcome this hurdle, three solutions are introduced to convert model realizations to a common dimensionality while maintaining the statistical characteristics of the ensemble. A scalar, a vector and a matrix representation for models is presented, inferred from tomographic subsurface investigations, and three classic convergence assessment techniques are applied on them. It is shown that appropriately chosen scalar conversions of the models could retain similar statistical ensemble properties as geologic projections created by rasterization. KW - transdimensional inversion KW - MCMC modelling KW - convergence assessment Y1 - 2019 U6 - https://doi.org/10.1007/s11004-019-09811-x SN - 1874-8961 SN - 1874-8953 VL - 52 IS - 5 SP - 651 EP - 668 PB - Springer CY - Heidelberg ER - TY - JOUR A1 - Ly, Ibrahim T1 - A Cauchy problem for the Cauchy-Riemann operator JF - Afrika Matematika N2 - We study the Cauchy problem for a nonlinear elliptic equation with data on a piece S of the boundary surface partial derivative X. By the Cauchy problem is meant any boundary value problem for an unknown function u in a domain X with the property that the data on S, if combined with the differential equations in X, allows one to determine all derivatives of u on S by means of functional equations. In the case of real analytic data of the Cauchy problem, the existence of a local solution near S is guaranteed by the Cauchy-Kovalevskaya theorem. We discuss a variational setting of the Cauchy problem which always possesses a generalized solution. KW - nonlinear PDI KW - Cauchy problem KW - Zaremba problem Y1 - 2020 U6 - https://doi.org/10.1007/s13370-020-00810-4 SN - 1012-9405 SN - 2190-7668 VL - 32 IS - 1-2 SP - 69 EP - 76 PB - Springer CY - Heidelberg ER - TY - JOUR A1 - Malass, Ihsane A1 - Tarkhanov, Nikolaj Nikolaevič T1 - A perturbation of the de Rham complex T1 - Возмущение комплекса де Рама JF - Journal of Siberian Federal University : Mathematics & Physics JF - Žurnal Sibirskogo Federalʹnogo Universiteta : Matematika i fizika N2 - We consider a perturbation of the de Rham complex on a compact manifold with boundary. This perturbation goes beyond the framework of complexes, and so cohomology does not apply to it. On the other hand, its curvature is "small", hence there is a natural way to introduce an Euler characteristic and develop a Lefschetz theory for the perturbation. This work is intended as an attempt to develop a cohomology theory for arbitrary sequences of linear mappings. N2 - Рассмотрим возмущение комплекса де Рама на компактном многообразии с краем. Это возмущение выходит за рамки комплексов, и поэтому когомологии к нему не относятся. С другой стороны, его кривизна "мала", поэтому существует естественный способ ввести характеристику Эйлера и разработать теорию Лефшеца для возмущения. Данная работа предназначена для попытки разработать теорию когомологий для произвольных последовательностей линейных отображений. KW - de Rham complex KW - cohomology KW - Hodge theory KW - Neumann problem KW - комплекс де Рама KW - когомологии KW - теория Ходжа KW - проблема Неймана Y1 - 2020 U6 - https://doi.org/10.17516/1997-1397-2020-13-5-519-532 SN - 1997-1397 SN - 2313-6022 VL - 13 IS - 5 SP - 519 EP - 532 PB - Siberian Federal University CY - Krasnojarsk ER - TY - JOUR A1 - Al-Saedy, Ammar Jaffar Muhesin A1 - Tarchanov, Nikolaj Nikolaevič T1 - A degree theory for Lagrangian boundary value problems JF - Žurnal Sibirskogo Federalʹnogo Universiteta = Journal of Siberian Federal University; mathematics & physics N2 - We study those nonlinear partial differential equations which appear as Euler-Lagrange equations of variational problems. On defining weak boundary values of solutions to such equations we initiate the theory of Lagrangian boundary value problems in spaces of appropriate smoothness. We also analyse if the concept of mapping degree of current importance applies to Lagrangian problems. N2 - Мы изучаем те нелинейные уравнения с частными производными, которые возникают как уравнения Эйлера-Лагранжа вариационных задач. Определяя слабые граничные значения решений таких уравнений, мы инициируем теорию лагранжевых краевых задач в функциональных пространствах подходящей гладкости. Мы также анализируем, применяется ли современная концепция степени отображения к лагранжевым проблемам. KW - nonlinear equations KW - Lagrangian system KW - weak boundary values KW - quasilinear Fredholm operators KW - mapping degree Y1 - 2020 U6 - https://doi.org/10.17516/1997-1397-2020-13-1-5-25 SN - 1997-1397 SN - 2313-6022 VL - 13 IS - 1 SP - 5 EP - 25 PB - Sibirskij Federalʹnyj Universitet CY - Krasnojarsk ER - TY - CHAP A1 - Clavier, Pierre J. A1 - Guo, Li A1 - Paycha, Sylvie A1 - Zhang, Bin T1 - Renormalisation and locality BT - branched zeta values T2 - Algebraic Combinatorics, Resurgence, Moulds and Applications (CARMA) Volume 2 Y1 - 2020 SN - 978-3-03719-205-4 print SN - 978-3-03719-705-9 online U6 - https://doi.org/10.4171/205 SP - 85 EP - 132 PB - European Mathematical Society Publishing House CY - Zürich ER - TY - JOUR A1 - Chelkh, W. A1 - Ly, Ibrahim A1 - Tarkhanov, Nikolai T1 - A remark on the Laplace transform JF - Siberian Mathematical Journal N2 - The study of the Cauchy problem for solutions of the heat equation in a cylindrical domain with data on the lateral surface by the Fourier method raises the problem of calculating the inverse Laplace transform of the entire function cos root z. This problem has no solution in the standard theory of the Laplace transform. We give an explicit formula for the inverse Laplace transform of cos root z using the theory of analytic functionals. This solution suits well to efficiently develop the regularization of solutions to Cauchy problems for parabolic equations with data on noncharacteristic surfaces. KW - Fourier-Laplace transform KW - distributions with one-sided support KW - holomorphic function KW - analytic functional Y1 - 2020 U6 - https://doi.org/10.1134/S0037446620040151 SN - 0037-4466 SN - 1573-9260 VL - 61 IS - 4 SP - 755 EP - 762 PB - Consultants Bureau, Springer CY - New York ER - TY - JOUR A1 - Keller, Matthias A1 - Schwarz, Michael T1 - Courant’s nodal domain theorem for positivity preserving forms JF - Journal of spectral theory N2 - We introduce a notion of nodal domains for positivity preserving forms. This notion generalizes the classical ones for Laplacians on domains and on graphs. We prove the Courant nodal domain theorem in this generalized setting using purely analytical methods. KW - Nodal domain KW - eigenfunction KW - Dirichlet form KW - compact resolvent Y1 - 2020 U6 - https://doi.org/10.4171/JST/292 SN - 1664-039X SN - 1664-0403 VL - 10 IS - 1 SP - 271 EP - 309 PB - EMS Publishing House CY - Zürich ER - TY - JOUR A1 - Ly, Ibrahim A1 - Tarkhanov, Nikolaj Nikolaevič T1 - Asymptotic expansions at nonsymmetric cuspidal points JF - Mathematical notes N2 - We study the asymptotics of solutions to the Dirichlet problem in a domain X subset of R3 whose boundary contains a singular point O. In a small neighborhood of this point, the domain has the form {z > root x(2) + y(4)}, i.e., the origin is a nonsymmetric conical point at the boundary. So far, the behavior of solutions to elliptic boundary-value problems has not been studied sufficiently in the case of nonsymmetric singular points. This problem was posed by V.A. Kondrat'ev in 2000. We establish a complete asymptotic expansion of solutions near the singular point. KW - Dirichlet problem KW - singular points KW - asymptotic expansions Y1 - 2020 U6 - https://doi.org/10.1134/S0001434620070238 SN - 0001-4346 SN - 1573-8876 VL - 108 IS - 1-2 SP - 219 EP - 228 PB - Springer Science CY - New York ER -