1,403 research outputs found
A Sufficient Condition for Graphic Sequences with Given Largest and Smallest Entries, Length, and Sum
We give a sufficient condition for a degree sequence to be graphic based on
its largest and smallest elements, length, and sum. This bound generalizes a
result of Zverovich and Zverovich
A sufficient condition for a pair of sequences to be bipartite graphic
We present a sufficient condition for a pair of finite integer sequences to
be degree sequences of a bipartite graph, based only on the lengths of the
sequences and their largest and smallest elements.Comment: 5 page
Symmetric Bipartite Graphs and Graphs with Loops
We show that if the two parts of a finite bipartite graph have the same
degree sequence, then there is a bipartite graph, with the same degree
sequences, which is symmetric, in that it has an involutive graph automorphism
that interchanges its two parts. To prove this, we study the relationship
between symmetric bipartite graphs and graphs with loops.Comment: arXiv admin note: substantial text overlap with arXiv:1302.365
AFLOW-SYM: Platform for the complete, automatic and self-consistent symmetry analysis of crystals
Determination of the symmetry profile of structures is a persistent challenge
in materials science. Results often vary amongst standard packages, hindering
autonomous materials development by requiring continuous user attention and
educated guesses. Here, we present a robust procedure for evaluating the
complete suite of symmetry properties, featuring various representations for
the point-, factor-, space groups, site symmetries, and Wyckoff positions. The
protocol determines a system-specific mapping tolerance that yields symmetry
operations entirely commensurate with fundamental crystallographic principles.
The self consistent tolerance characterizes the effective spatial resolution of
the reported atomic positions. The approach is compared with the most used
programs and is successfully validated against the space group information
provided for over 54,000 entries in the Inorganic Crystal Structure Database.
Subsequently, a complete symmetry analysis is applied to all 1.7 million
entries of the AFLOW data repository. The AFLOW-SYM package has been
implemented in, and made available for, public use through the automated,
framework AFLOW.Comment: 24 pages, 6 figure
Optimal prefix codes for pairs of geometrically-distributed random variables
Optimal prefix codes are studied for pairs of independent, integer-valued
symbols emitted by a source with a geometric probability distribution of
parameter , . By encoding pairs of symbols, it is possible to
reduce the redundancy penalty of symbol-by-symbol encoding, while preserving
the simplicity of the encoding and decoding procedures typical of Golomb codes
and their variants. It is shown that optimal codes for these so-called
two-dimensional geometric distributions are \emph{singular}, in the sense that
a prefix code that is optimal for one value of the parameter cannot be
optimal for any other value of . This is in sharp contrast to the
one-dimensional case, where codes are optimal for positive-length intervals of
the parameter . Thus, in the two-dimensional case, it is infeasible to give
a compact characterization of optimal codes for all values of the parameter
, as was done in the one-dimensional case. Instead, optimal codes are
characterized for a discrete sequence of values of that provide good
coverage of the unit interval. Specifically, optimal prefix codes are described
for (), covering the range , and
(), covering the range . The described codes produce the expected
reduction in redundancy with respect to the one-dimensional case, while
maintaining low complexity coding operations.Comment: To appear in IEEE Transactions on Information Theor
AFLOW-SYM: platform for the complete, automatic and self-consistent symmetry analysis of crystals
Determination of the symmetry profile of structures is a persistent challenge in materials science. Results often vary amongst standard packages, hindering autonomous materials development by requiring continuous user attention and educated guesses. This article presents a robust procedure for evaluating the complete suite of symmetry properties, featuring various representations for the point, factor and space groups, site symmetries and Wyckoff positions. The protocol determines a system-specific mapping tolerance that yields symmetry operations entirely commensurate with fundamental crystallographic principles. The self-consistent tolerance characterizes the effective spatial resolution of the reported atomic positions. The approach is compared with the most used programs and is successfully validated against the space-group information provided for over 54 000 entries in the Inorganic Crystal Structure Database (ICSD). Subsequently, a complete symmetry analysis is applied to all 1.7+ million entries of the AFLOW data repository. The AFLOW-SYM package has been implemented in, and made available for, public use through the automated ab initio framework AFLOW
Symbolic and Visual Retrieval of Mathematical Notation using Formula Graph Symbol Pair Matching and Structural Alignment
Large data collections containing millions of math formulae in different formats are available on-line. Retrieving math expressions from these collections is challenging. We propose a framework for retrieval of mathematical notation using symbol pairs extracted from visual and semantic representations of mathematical expressions on the symbolic domain for retrieval of text documents. We further adapt our model for retrieval of mathematical notation on images and lecture videos. Graph-based representations are used on each modality to describe math formulas. For symbolic formula retrieval, where the structure is known, we use symbol layout trees and operator trees. For image-based formula retrieval, since the structure is unknown we use a more general Line of Sight graph representation. Paths of these graphs define symbol pairs tuples that are used as the entries for our inverted index of mathematical notation. Our retrieval framework uses a three-stage approach with a fast selection of candidates as the first layer, a more detailed matching algorithm with similarity metric computation in the second stage, and finally when relevance assessments are available, we use an optional third layer with linear regression for estimation of relevance using multiple similarity scores for final re-ranking. Our model has been evaluated using large collections of documents, and preliminary results are presented for videos and cross-modal search. The proposed framework can be adapted for other domains like chemistry or technical diagrams where two visually similar elements from a collection are usually related to each other
Efficient Point-Cloud Processing with Primitive Shapes
This thesis presents methods for efficient processing of point-clouds based on primitive shapes. The set of considered simple parametric shapes consists of planes, spheres, cylinders, cones and tori. The algorithms developed in this work are targeted at scenarios in which the occurring surfaces can be well represented by this set of shape primitives which is the case in many man-made environments such as e.g. industrial compounds, cities or building interiors. A primitive subsumes a set of corresponding points in the point-cloud and serves as a proxy for them. Therefore primitives are well suited to directly address the unavoidable oversampling of large point-clouds and lay the foundation for efficient point-cloud processing algorithms. The first contribution of this thesis is a novel shape primitive detection method that is efficient even on very large and noisy point-clouds. Several applications for the detected primitives are subsequently explored, resulting in a set of novel algorithms for primitive-based point-cloud processing in the areas of compression, recognition and completion. Each of these application directly exploits and benefits from one or more of the detected primitives' properties such as approximation, abstraction, segmentation and continuability
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