15,198 research outputs found
Solving systems of transcendental equations involving the Heun functions
The Heun functions have wide application in modern physics and are expected
to succeed the hypergeometrical functions in the physical problems of the 21st
century. The numerical work with those functions, however, is complicated and
requires filling the gaps in the theory of the Heun functions and also,
creating new algorithms able to work with them efficiently.
We propose a new algorithm for solving a system of two nonlinear
transcendental equations with two complex variables based on the M\"uller
algorithm. The new algorithm is particularly useful in systems featuring the
Heun functions and for them, the new algorithm gives distinctly better results
than Newton's and Broyden's methods.
As an example for its application in physics, the new algorithm was used to
find the quasi-normal modes (QNM) of Schwarzschild black hole described by the
Regge-Wheeler equation. The numerical results obtained by our method are
compared with the already published QNM frequencies and are found to coincide
to a great extent with them. Also discussed are the QNM of the Kerr black hole,
described by the Teukolsky Master equation.Comment: 17 pages, 4 figures. Typos corrected, one figure added, some sections
revised. The article is a rework of the internal report arXiv:1005.537
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A Framework for Globally Optimizing Mixed-Integer Signomial Programs
Mixed-integer signomial optimization problems have broad applicability in engineering. Extending the Global Mixed-Integer Quadratic Optimizer, GloMIQO (Misener, Floudas in J. Glob. Optim., 2012. doi:10.1007/s10898-012-9874-7), this manuscript documents a computational framework for deterministically addressing mixed-integer signomial optimization problems to ε-global optimality. This framework generalizes the GloMIQO strategies of (1) reformulating user input, (2) detecting special mathematical structure, and (3) globally optimizing the mixed-integer nonconvex program. Novel contributions of this paper include: flattening an expression tree towards term-based data structures; introducing additional nonconvex terms to interlink expressions; integrating a dynamic implementation of the reformulation-linearization technique into the branch-and-cut tree; designing term-based underestimators that specialize relaxation strategies according to variable bounds in the current tree node. Computational results are presented along with comparison of the computational framework to several state-of-the-art solvers. © 2013 Springer Science+Business Media New York
A Cycle-Based Formulation and Valid Inequalities for DC Power Transmission Problems with Switching
It is well-known that optimizing network topology by switching on and off
transmission lines improves the efficiency of power delivery in electrical
networks. In fact, the USA Energy Policy Act of 2005 (Section 1223) states that
the U.S. should "encourage, as appropriate, the deployment of advanced
transmission technologies" including "optimized transmission line
configurations". As such, many authors have studied the problem of determining
an optimal set of transmission lines to switch off to minimize the cost of
meeting a given power demand under the direct current (DC) model of power flow.
This problem is known in the literature as the Direct-Current Optimal
Transmission Switching Problem (DC-OTS). Most research on DC-OTS has focused on
heuristic algorithms for generating quality solutions or on the application of
DC-OTS to crucial operational and strategic problems such as contingency
correction, real-time dispatch, and transmission expansion. The mathematical
theory of the DC-OTS problem is less well-developed. In this work, we formally
establish that DC-OTS is NP-Hard, even if the power network is a
series-parallel graph with at most one load/demand pair. Inspired by Kirchoff's
Voltage Law, we give a cycle-based formulation for DC-OTS, and we use the new
formulation to build a cycle-induced relaxation. We characterize the convex
hull of the cycle-induced relaxation, and the characterization provides strong
valid inequalities that can be used in a cutting-plane approach to solve the
DC-OTS. We give details of a practical implementation, and we show promising
computational results on standard benchmark instances
On the minimum orbital intersection distance computation: a new effective method
The computation of the Minimum Orbital Intersection Distance (MOID) is an
old, but increasingly relevant problem. Fast and precise methods for MOID
computation are needed to select potentially hazardous asteroids from a large
catalogue. The same applies to debris with respect to spacecraft. An iterative
method that strictly meets these two premises is presented.Comment: 13 pages, 10 figures, article accepted for publication in MNRA
Towards Verifying Nonlinear Integer Arithmetic
We eliminate a key roadblock to efficient verification of nonlinear integer
arithmetic using CDCL SAT solvers, by showing how to construct short resolution
proofs for many properties of the most widely used multiplier circuits. Such
short proofs were conjectured not to exist. More precisely, we give n^{O(1)}
size regular resolution proofs for arbitrary degree 2 identities on array,
diagonal, and Booth multipliers and quasipolynomial- n^{O(\log n)} size proofs
for these identities on Wallace tree multipliers.Comment: Expanded and simplified with improved result
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