SP, a Package for Schubert Polynomials Realized with the Computer Algebra System MAPLE

AbstractWe present theSPpackage devoted to the manipulation of Schubert polynomials. These polynomials contain as a subfamily the Schur symmetric functions and allow to extend to non symmetric polynomials the classical combinatorial techniques of the theory of symmetric functions. They have many applications, ranging from multivariate interpolation to intersection theory in algebraic geometry

Computation and Physics in Algebraic Geometry

Physics provides new, tantalizing problems that we solve by developing and implementing innovative and effective geometric tools in nonlinear algebra. The techniques we employ also rely on numerical and symbolic computations performed with computer algebra. First, we study solutions to the Kadomtsev-Petviashvili equation that arise from singular curves. The Kadomtsev-Petviashvili equation is a partial differential equation describing nonlinear wave motion whose solutions can be built from an algebraic curve. Such a surprising connection established by Krichever and Shiota also led to an entirely new point of view on a classical problem in algebraic geometry known as the Schottky problem. To explore the connection with curves with at worst nodal singularities, we define the Hirota variety, which parameterizes KP solutions arising from such curves. Studying the geometry of the Hirota variety provides a new approach to the Schottky problem. We investigate it for irreducible rational nodal curves, giving a partial solution to the weak Schottky problem in this case. Second, we formulate questions from scattering amplitudes in a broader context using very affine varieties and D-module theory. The interplay between geometry and combinatorics in particle physics indeed suggests an underlying, coherent mathematical structure behind the study of particle interactions. In this thesis, we gain a better understanding of mathematical objects, such as moduli spaces of point configurations and generalized Euler integrals, for which particle physics provides concrete, non-trivial examples, and we prove some conjectures stated in the physics literature. Finally, we study linear spaces of symmetric matrices, addressing questions motivated by algebraic statistics, optimization, and enumerative geometry. This includes giving explicit formulas for the maximum likelihood degree and studying tangency problems for quadric surfaces in projective space from the point of view of real algebraic geometry

The Pauli principle eevisited

By the Pauli exclusion principle, no quantum state can be occupied by more than one electron. One can state this as a constraint on the one electron density matrix that bounds its eigenvalues by 1. Shortly after its discovery, the Pauli principle was replaced by anti-symmetry of the multi-electron wave function. In this paper we solve a longstanding problem about the impact of this replacement on the one electron density matrix, that goes far beyond the original Pauli principle. Our approach uses Berenstein and Sjamaar's theorem on the restriction of an adjoint orbit onto a subgroup, and allows us to treat any type of permutational symmetry. © 2008 Springer-Verlag

The Pauli principe, representation theory, and geometry of flag varieties

Ankara : The Department of Mathematics and the Institute of Engineering and Sciences of Bilkent University, 2008.Thesis (Ph.D.) -- Bilkent University, 2008.Includes bibliographical references leaves 113-116.According to the Pauli exclusion principle, discovered in 1925, no two identical electrons may occupy the same quantum state. In terms of electron density matrix this amounts to an upper bound for its eigenvalues by 1. In 1926, it has been replaced by skew-symmetry of a multi-electron wave function. In this thesis we give two different solutions to a problem about the impact of this replacement on the electron density matrix, which goes far beyond the original Pauli principle.Altunbulak, MuratPh.D

Internationales Kolloquium über Anwendungen der Informatik und Mathematik in Architektur und Bauwesen : 04. bis 06.07. 2012, Bauhaus-Universität Weimar

The 19th International Conference on the Applications of Computer Science and Mathematics in Architecture and Civil Engineering will be held at the Bauhaus University Weimar from 4th till 6th July 2012. Architects, computer scientists, mathematicians, and engineers from all over the world will meet in Weimar for an interdisciplinary exchange of experiences, to report on their results in research, development and practice and to discuss. The conference covers a broad range of research areas: numerical analysis, function theoretic methods, partial differential equations, continuum mechanics, engineering applications, coupled problems, computer sciences, and related topics. Several plenary lectures in aforementioned areas will take place during the conference. We invite architects, engineers, designers, computer scientists, mathematicians, planners, project managers, and software developers from business, science and research to participate in the conference

Internationales Kolloquium über Anwendungen der Informatik und Mathematik in Architektur und Bauwesen : 04. bis 06.07. 2012, Bauhaus-Universität Weimar

The 19th International Conference on the Applications of Computer Science and Mathematics in Architecture and Civil Engineering will be held at the Bauhaus University Weimar from 4th till 6th July 2012. Architects, computer scientists, mathematicians, and engineers from all over the world will meet in Weimar for an interdisciplinary exchange of experiences, to report on their results in research, development and practice and to discuss. The conference covers a broad range of research areas: numerical analysis, function theoretic methods, partial differential equations, continuum mechanics, engineering applications, coupled problems, computer sciences, and related topics. Several plenary lectures in aforementioned areas will take place during the conference. We invite architects, engineers, designers, computer scientists, mathematicians, planners, project managers, and software developers from business, science and research to participate in the conference

Exploring potential energy surfaces in ground- and excited states

Chemical reactivity of atoms, molecules and ions is governed by their underlying potential energy surface. Calculating the whole potential energy surface within reasonable bounds, is impossible for all but the smallest molecules. Usually, only parts of the full potential energy surface can be studied, namely stationary points and the minimum energy paths connecting them. By comparing energies of stationary points and their separating barriers, conclusions regarding possible reactions mechanism, or their infeasibility, can be drawn. Taking excited states into account leads to further complications, as now multiple potential energy surfaces have to be considered and root flips between different excited states may occur, requiring effective state-tracking. Part II of this thesis describes the required methods to locate stationary points and minimum energy paths on potential energy surfaces, by using surface-walking, chain-of-states optimization and intrinsic reaction coordinate integration. Several approaches to state-tracking are presented in chapter 4. Results of this thesis are presented in Part III, containing two contributions to the field of photochemistry: chapter 12 provides a possible excited-state reaction mechanism for a biaryl cross-coupling reaction and offers a plausible explanation for its high regioselectivity. The second contribution is the development pysisyphus (chapter 13), an external optimizer implemented in python, aware of excited states and thus the core of this thesis. By implementing the state-tracking algorithms outlined in chapter 4 it allows effective and efficient optimizations of stationary points in ground- and excited-states. The performance of pysisyphus is verified for several established benchmark sets. Results for several excited-state optimizations are presented in section 13.3, where pysisyphus shows good performance for the optimization of sizeable transition-metal complexes

Software for Exascale Computing - SPPEXA 2016-2019

This open access book summarizes the research done and results obtained in the second funding phase of the Priority Program 1648 "Software for Exascale Computing" (SPPEXA) of the German Research Foundation (DFG) presented at the SPPEXA Symposium in Dresden during October 21-23, 2019. In that respect, it both represents a continuation of Vol. 113 in Springer’s series Lecture Notes in Computational Science and Engineering, the corresponding report of SPPEXA’s first funding phase, and provides an overview of SPPEXA’s contributions towards exascale computing in today's sumpercomputer technology. The individual chapters address one or more of the research directions (1) computational algorithms, (2) system software, (3) application software, (4) data management and exploration, (5) programming, and (6) software tools. The book has an interdisciplinary appeal: scholars from computational sub-fields in computer science, mathematics, physics, or engineering will find it of particular interest