136,244 research outputs found
Rigorous Multiple-Precision Evaluation of D-Finite Functions in SageMath
We present a new open source implementation in the SageMath computer algebra
system of algorithms for the numerical solution of linear ODEs with polynomial
coefficients. Our code supports regular singular connection problems and
provides rigorous error bounds
Formal Theories for Linear Algebra
We introduce two-sorted theories in the style of [CN10] for the complexity
classes \oplusL and DET, whose complete problems include determinants over Z2
and Z, respectively. We then describe interpretations of Soltys' linear algebra
theory LAp over arbitrary integral domains, into each of our new theories. The
result shows equivalences of standard theorems of linear algebra over Z2 and Z
can be proved in the corresponding theory, but leaves open the interesting
question of whether the theorems themselves can be proved.Comment: This is a revised journal version of the paper "Formal Theories for
Linear Algebra" (Computer Science Logic) for the journal Logical Methods in
Computer Scienc
Every generalized quadrangle of order 5 having a regular point is symplectic
For many years now, one of the most important open problems in the theory of generalized quadrangles has been whether other classes of generalized quadrangles exist besides those that are currently known. This paper reports on an unsuccessful attempt to construct a new generalized quadrangle. As a byproduct of our attempt, however, we obtain the following new characterization result: every generalized quadrangle of order 5 that has at least one regular point is isomorphic to the quadrangle W(5) arising from a symplectic polarity of PG(3, 5). During the classification process, we used the computer algebra system GAP to perform certain computations or to search for an optimal strategy for the proof
Governing Singularities of Schubert Varieties
We present a combinatorial and computational commutative algebra methodology
for studying singularities of Schubert varieties of flag manifolds.
We define the combinatorial notion of *interval pattern avoidance*. For
"reasonable" invariants P of singularities, we geometrically prove that this
governs (1) the P-locus of a Schubert variety, and (2) which Schubert varieties
are globally not P. The prototypical case is P="singular"; classical pattern
avoidance applies admirably for this choice [Lakshmibai-Sandhya'90], but is
insufficient in general.
Our approach is analyzed for some common invariants, including
Kazhdan-Lusztig polynomials, multiplicity, factoriality, and Gorensteinness,
extending [Woo-Yong'05]; the description of the singular locus (which was
independently proved by [Billey-Warrington '03], [Cortez '03],
[Kassel-Lascoux-Reutenauer'03], [Manivel'01]) is also thus reinterpreted.
Our methods are amenable to computer experimentation, based on computing with
*Kazhdan-Lusztig ideals* (a class of generalized determinantal ideals) using
Macaulay 2. This feature is supplemented by a collection of open problems and
conjectures.Comment: 23 pages. Software available at the authors' webpages. Version 2 is
the submitted version. It has a nomenclature change: "Bruhat-restricted
pattern avoidance" is renamed "interval pattern avoidance"; the introduction
has been reorganize
Light on the Infinite Group Relaxation
This is a survey on the infinite group problem, an infinite-dimensional
relaxation of integer linear optimization problems introduced by Ralph Gomory
and Ellis Johnson in their groundbreaking papers titled "Some continuous
functions related to corner polyhedra I, II" [Math. Programming 3 (1972),
23-85, 359-389]. The survey presents the infinite group problem in the modern
context of cut generating functions. It focuses on the recent developments,
such as algorithms for testing extremality and breakthroughs for the k-row
problem for general k >= 1 that extend previous work on the single-row and
two-row problems. The survey also includes some previously unpublished results;
among other things, it unveils piecewise linear extreme functions with more
than four different slopes. An interactive companion program, implemented in
the open-source computer algebra package Sage, provides an updated compendium
of known extreme functions.Comment: 45 page
Computational Geometric and Algebraic Topology
Computational topology is a young, emerging field of mathematics that seeks out practical algorithmic methods for solving complex and fundamental problems in geometry and topology. It draws on a wide variety of techniques from across pure mathematics (including topology, differential geometry, combinatorics, algebra, and discrete geometry), as well as applied mathematics and theoretical computer science. In turn, solutions to these problems have a wide-ranging impact: already they have enabled significant progress in the core area of geometric topology, introduced new methods in applied mathematics, and yielded new insights into the role that topology has to play in fundamental problems surrounding computational complexity.
At least three significant branches have emerged in computational topology: algorithmic 3-manifold and knot theory, persistent homology and surfaces and graph embeddings. These branches have emerged largely independently. However, it is clear that they have much to offer each other. The goal of this workshop was to be the first significant step to bring these three areas together, to share ideas in depth, and to pool our expertise in approaching some of the major open problems in the field
Robust Computer Algebra, Theorem Proving, and Oracle AI
In the context of superintelligent AI systems, the term "oracle" has two
meanings. One refers to modular systems queried for domain-specific tasks.
Another usage, referring to a class of systems which may be useful for
addressing the value alignment and AI control problems, is a superintelligent
AI system that only answers questions. The aim of this manuscript is to survey
contemporary research problems related to oracles which align with long-term
research goals of AI safety. We examine existing question answering systems and
argue that their high degree of architectural heterogeneity makes them poor
candidates for rigorous analysis as oracles. On the other hand, we identify
computer algebra systems (CASs) as being primitive examples of domain-specific
oracles for mathematics and argue that efforts to integrate computer algebra
systems with theorem provers, systems which have largely been developed
independent of one another, provide a concrete set of problems related to the
notion of provable safety that has emerged in the AI safety community. We
review approaches to interfacing CASs with theorem provers, describe
well-defined architectural deficiencies that have been identified with CASs,
and suggest possible lines of research and practical software projects for
scientists interested in AI safety.Comment: 15 pages, 3 figure
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