104,864 research outputs found
Clifford analysis between continuous and discrete
Some decades ago D. Knuth et al. have coined concrete mathematics as the blending of CONtinuous and
disCRETE math, taking into account that problems of standard discrete mathematics can often be solved by
methods based on continuous mathematics together with a controlled manipulation of mathematical formulas.
Of course, it was not a new idea, but due to the ongoing emergence of computer aided algebraic manipulation
tools of that time it emphasized their use for elegant solutions of old problems or even the detection of new
important relationships. Our aim is to show that the same philosophy can be successfully applied to Clifford
Analysis by taking advantages of its inherent non-commutative algebra to obtain results or develop methods
that are di erent from other ones. In particular, we determine new binomial sums by using a hypercomplex
generating function for a special type of monogenic polynomials and develop an algorithm for the determination
of their scalar and vector part which illustrates well the diifferences to the corresponding complex case.The research of the first author was partially supported by the R&D Unit Matemdtica e Aplicagoes (UIMA) of the University of Aveiro, through the Portuguese Foundation for Science and Technology (FCT)
Curriculum Guidelines for Undergraduate Programs in Data Science
The Park City Math Institute (PCMI) 2016 Summer Undergraduate Faculty Program
met for the purpose of composing guidelines for undergraduate programs in Data
Science. The group consisted of 25 undergraduate faculty from a variety of
institutions in the U.S., primarily from the disciplines of mathematics,
statistics and computer science. These guidelines are meant to provide some
structure for institutions planning for or revising a major in Data Science
Multi-layer Architecture For Storing Visual Data Based on WCF and Microsoft SQL Server Database
In this paper we present a novel architecture for storing visual data.
Effective storing, browsing and searching collections of images is one of the
most important challenges of computer science. The design of architecture for
storing such data requires a set of tools and frameworks such as SQL database
management systems and service-oriented frameworks. The proposed solution is
based on a multi-layer architecture, which allows to replace any component
without recompilation of other components. The approach contains five
components, i.e. Model, Base Engine, Concrete Engine, CBIR service and
Presentation. They were based on two well-known design patterns: Dependency
Injection and Inverse of Control. For experimental purposes we implemented the
SURF local interest point detector as a feature extractor and -means
clustering as indexer. The presented architecture is intended for content-based
retrieval systems simulation purposes as well as for real-world CBIR tasks.Comment: Accepted for the 14th International Conference on Artificial
Intelligence and Soft Computing, ICAISC, June 14-18, 2015, Zakopane, Polan
The MMT API: A Generic MKM System
The MMT language has been developed as a scalable representation and
interchange language for formal mathematical knowledge. It permits natural
representations of the syntax and semantics of virtually all declarative
languages while making MMT-based MKM services easy to implement. It is
foundationally unconstrained and can be instantiated with specific formal
languages.
The MMT API implements the MMT language along with multiple backends for
persistent storage and frontends for machine and user access. Moreover, it
implements a wide variety of MMT-based knowledge management services. The API
and all services are generic and can be applied to any language represented in
MMT. A plugin interface permits injecting syntactic and semantic idiosyncrasies
of individual formal languages.Comment: Conferences on Intelligent Computer Mathematics (CICM) 2013 The final
publication is available at http://link.springer.com
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
Towards MKM in the Large: Modular Representation and Scalable Software Architecture
MKM has been defined as the quest for technologies to manage mathematical
knowledge. MKM "in the small" is well-studied, so the real problem is to scale
up to large, highly interconnected corpora: "MKM in the large". We contend that
advances in two areas are needed to reach this goal. We need representation
languages that support incremental processing of all primitive MKM operations,
and we need software architectures and implementations that implement these
operations scalably on large knowledge bases.
We present instances of both in this paper: the MMT framework for modular
theory-graphs that integrates meta-logical foundations, which forms the base of
the next OMDoc version; and TNTBase, a versioned storage system for XML-based
document formats. TNTBase becomes an MMT database by instantiating it with
special MKM operations for MMT.Comment: To appear in The 9th International Conference on Mathematical
Knowledge Management: MKM 201
An algorithmic approach to the existence of ideal objects in commutative algebra
The existence of ideal objects, such as maximal ideals in nonzero rings,
plays a crucial role in commutative algebra. These are typically justified
using Zorn's lemma, and thus pose a challenge from a computational point of
view. Giving a constructive meaning to ideal objects is a problem which dates
back to Hilbert's program, and today is still a central theme in the area of
dynamical algebra, which focuses on the elimination of ideal objects via
syntactic methods. In this paper, we take an alternative approach based on
Kreisel's no counterexample interpretation and sequential algorithms. We first
give a computational interpretation to an abstract maximality principle in the
countable setting via an intuitive, state based algorithm. We then carry out a
concrete case study, in which we give an algorithmic account of the result that
in any commutative ring, the intersection of all prime ideals is contained in
its nilradical
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