analytic combinatorics in several variables refers to a suite of tools that provide sharp asymptotic estimates for certain combinatorial quantities. In this paper, we apply these tools to determine the Gilbert-Varsham...
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analytic combinatorics in several variables refers to a suite of tools that provide sharp asymptotic estimates for certain combinatorial quantities. In this paper, we apply these tools to determine the Gilbert-Varshamov lower bound on the rate of optimal codes in L-1 metric. Several different code spaces are analyzed, including the simplex and the hypercube in Z(n) , all of which are inspired by concrete data storage and transmission models such as the permutation channel, the repetition channel, the adjacent transposition (bit-shift) channel, the multilevel flash memory channel, etc.
Using methods from analytic combinatorics, we study the families of perfect matchings, partitions, chord diagrams, and hyperchord diagrams on a disk with a prescribed number of crossings. For each family, we express t...
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Using methods from analytic combinatorics, we study the families of perfect matchings, partitions, chord diagrams, and hyperchord diagrams on a disk with a prescribed number of crossings. For each family, we express the generating function of the configurations with exactly k crossings as a rational function of the generating function of crossing-free configurations. Using these expressions, we study the singular behavior of these generating functions and derive asymptotic results on the counting sequences of the configurations with precisely k crossings. Limiting distributions and random generators are also studied. (C) 2014 Elsevier Inc. All rights reserved.
This paper is intended to give for a general mathematical audience (including non-logicians) a survey of intriguing connections between analytic combinatorics and logic. We define the ordinals below epsilon(0) in non-...
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This paper is intended to give for a general mathematical audience (including non-logicians) a survey of intriguing connections between analytic combinatorics and logic. We define the ordinals below epsilon(0) in non-logical terms and we survey a selection of recent results about the analytic combinatorics of these ordinals. Using a versatile and flexible (logarithmic) compression technique we give applications to phase transitions for independence results, Hilbert's basis theorem, local number theory, Ramsey theory, Hydra games, and Goodstein sequences. We discuss briefly universality and renormalization issues in this context. Finally, we indicate how regularity properties of ordinal count functions can be used to prove logical limit laws. (c) 2005 Elsevier B.V. All rights reserved.
We enumerate the connected graphs that contain a number of edges growing linearly with respect to the number of vertices. So far, only the first term of the asymptotics and a bound on the error were known. Using analy...
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We enumerate the connected graphs that contain a number of edges growing linearly with respect to the number of vertices. So far, only the first term of the asymptotics and a bound on the error were known. Using analytic combinatorics, that is, generating function manipulations, we derive a formula for the coefficients of the complete asymptotic expansion. The same result is derived for connected multigraphs.
The method of analytic combinatorics (AC) is a unified approach to multiple object tracking that encodes joint probability distributions into probability generating functionals (PGFLs). PGFLs characterize distribution...
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ISBN:
(纸本)9781538631034
The method of analytic combinatorics (AC) is a unified approach to multiple object tracking that encodes joint probability distributions into probability generating functionals (PGFLs). PGFLs characterize distributions exactly. A high level view of the tracking applications of PGFLs is outlined in this paper. Assignment models in well-known filters are modeled as products of PGFLs. MHT and multiBernoulli PGFLs are compared. Track extraction and the "notched" filter of the (reduced) Palm process are discussed. Bounded complexity approximate particle filter weights are found by saddle point methods applied to the Cauchy integral form of the derivatives.
The method of analytic combinatorics and labeling is shown to be a unifying framework in which to pose both high and low level data fusion problems. The method uses labeled generating functions. Several examples from ...
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ISBN:
(纸本)9780996452762
The method of analytic combinatorics and labeling is shown to be a unifying framework in which to pose both high and low level data fusion problems. The method uses labeled generating functions. Several examples from high level fusion and multiple target tracking are given. Examples from high level fusion include natural language processing and noisy graph association problems. Examples from multitarget tracking include multidimensional assignment problems, unlabeled and labeled JPDA, labeled multiBernoulli filters, and multihypothesis tracking.
On complex algebraic varieties, height functions arising in combinatorial applications fail to be proper. This complicates both the description and computation via Morse theory of key topological invariants. Here we e...
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On complex algebraic varieties, height functions arising in combinatorial applications fail to be proper. This complicates both the description and computation via Morse theory of key topological invariants. Here we establish checkable conditions under which the behavior at infinity may be ignored, and the usual theorems of classical and stratified Morse theory may be applied. This allows for simplified arguments in the field of analytic combinatorics in several variables, and forms the basis for new methods applying to problems beyond the reach of previous techniques.
This paper develops a unified enumerative and asymptotic theory of directed two-dimensional lattice paths in half-planes and quarter-planes. The lattice paths are specified by a finite set of rules that are both time ...
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This paper develops a unified enumerative and asymptotic theory of directed two-dimensional lattice paths in half-planes and quarter-planes. The lattice paths are specified by a finite set of rules that are both time and space homogeneous, and have a privileged direction of increase. (They are then essentially one-dimensional objects.) The theory relies on a specific "kernel method" that provides an important decomposition of the algebraic generating functions involved, as well as on a generic study of singularities of an associated algebraic curve. Consequences are precise computable estimates for the number of lattice paths of a given length under various constraints (bridges, excursions, meanders) as well as a characterization of the limit laws associated to several basic parameters of paths. (C) 2002 Elsevier Science B.V. All rights reserved.
Nowadays, increasing attention is being given to the study of the descriptional complexity in the average case. Although the underlying theory for such a study seems intimidating, one can obtain interesting results in...
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Nowadays, increasing attention is being given to the study of the descriptional complexity in the average case. Although the underlying theory for such a study seems intimidating, one can obtain interesting results in this area without too much effort. In this gentle introduction we take the reader on a journey through the basic analytical tools of that theory, giving some illustrative examples using regular expressions. Additionally, new asymptotic average-case results for several epsilon-NFA constructions are presented, in a unified framework. It turns out that, asymptotically, and in the average case, the complexity gap between the several constructions is significantly larger than in the worst case. Furthermore, one of the epsilon-NFA constructions approaches the corresponding epsilon-free NFA construction, asymptotically and on average. (C) 2014 Elsevier B.V. All rights reserved.
The partial derivative automaton (A(pd)) is usually smaller than other nondeterministic finite automata constructed from a regular expression, and it can be seen as a quotient of the Glushkov automaton (A(pos)). By es...
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The partial derivative automaton (A(pd)) is usually smaller than other nondeterministic finite automata constructed from a regular expression, and it can be seen as a quotient of the Glushkov automaton (A(pos)). By estimating the number of regular expressions that have E as a partial derivative, we compute a lower bound of the average number of mergings of states in A(pos) and describe its asymptotic behaviour. This depends on the alphabet size, k, and for growing k's its limit approaches half the number of states in A(pos). The lower bound corresponds to consider the A(pd) automaton for the marked version of the regular expression, i.e. where all its letters are made different. Experimental results suggest that the average number of states of this automaton, and of the A(pd) automaton for the unmarked regular expression, are very close to each other.
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