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12,080
Enumerative formulae for unrooted planar maps: a pattern
 ELECTRON J. COMBIN
, 2004
"... We present uniformly available simple enumerative formulae for unrooted planar nedge maps (counted up to orientationpreserving isomorphism) of numerous classes including arbitrary, loopless, nonseparable, eulerian maps and plane trees. All the formulae conform to a certain pattern with respect t ..."
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Cited by 5 (3 self)
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We present uniformly available simple enumerative formulae for unrooted planar nedge maps (counted up to orientationpreserving isomorphism) of numerous classes including arbitrary, loopless, nonseparable, eulerian maps and plane trees. All the formulae conform to a certain pattern with respect
Enumerative Formulae for Some Functions on Finite Sets
"... In the first section of this paper we give a matrix interpretation of wellknown inclusionexclusion principle. A result concerning (0, 1)matrices is also proved. In the next two section we apply these results to count the number of specified functions from a finite set into an another finite set. I ..."
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. In second section we first find a formula for the number of functions whose images contain a fixed subset. This gives a combinatorial interpretation for finite differences of the function n m. We also obtain an extension of the wellknown relation for Stirling numbers of the second kind. Two combinatorial
GromovWitten classes, quantum cohomology, and enumerative geometry
 Commun. Math. Phys
, 1994
"... The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov–Witten classes, and a discussion of their properties for Fano varieties. Cohomological ..."
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Cited by 484 (3 self)
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The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov–Witten classes, and a discussion of their properties for Fano varieties. Cohomological
Symbolic Model Checking for Realtime Systems
 INFORMATION AND COMPUTATION
, 1992
"... We describe finitestate programs over realnumbered time in a guardedcommand language with realvalued clocks or, equivalently, as finite automata with realvalued clocks. Model checking answers the question which states of a realtime program satisfy a branchingtime specification (given in an ..."
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Cited by 574 (50 self)
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We describe finitestate programs over realnumbered time in a guardedcommand language with realvalued clocks or, equivalently, as finite automata with realvalued clocks. Model checking answers the question which states of a realtime program satisfy a branchingtime specification (given in an extension of CTL with clock variables). We develop an algorithm that computes this set of states symbolically as a fixpoint of a functional on state predicates, without constructing the state space. For this purpose, we introduce a calculus on computation trees over realnumbered time. Unfortunately, many standard program properties, such as response for all nonzeno execution sequences (during which time diverges), cannot be characterized by fixpoints: we show that the expressiveness of the timed calculus is incomparable to the expressiveness of timed CTL. Fortunately, this result does not impair the symbolic verification of "implementable" realtime programsthose whose safety...
A theory of timed automata
, 1999
"... Model checking is emerging as a practical tool for automated debugging of complex reactive systems such as embedded controllers and network protocols (see [23] for a survey). Traditional techniques for model checking do not admit an explicit modeling of time, and are thus, unsuitable for analysis of ..."
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Cited by 2651 (32 self)
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Model checking is emerging as a practical tool for automated debugging of complex reactive systems such as embedded controllers and network protocols (see [23] for a survey). Traditional techniques for model checking do not admit an explicit modeling of time, and are thus, unsuitable for analysis of realtime systems whose correctness depends on relative magnitudes of different delays. Consequently, timed automata [7] were introduced as a formal notation to model the behavior of realtime systems. Its definition provides a simple way to annotate statetransition graphs with timing constraints using finitely many realvalued clock variables. Automated analysis of timed automata relies on the construction of a finite quotient of the infinite space of clock valuations. Over the years, the formalism has been extensively studied leading to many results establishing connections to circuits and logic, and much progress has been made in developing verification algorithms, heuristics, and tools. This paper provides a survey of the theory of timed automata, and their role in specification and verification of realtime systems.
The C Programming Language
, 1988
"... The C programming language was devised in the early 1970s as a system implementation language for the nascent Unix operating system. Derived from the typeless language BCPL, it evolved a type structure; created on a tiny machine as a tool to improve a meager programming environment, it has become on ..."
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Cited by 1527 (16 self)
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The C programming language was devised in the early 1970s as a system implementation language for the nascent Unix operating system. Derived from the typeless language BCPL, it evolved a type structure; created on a tiny machine as a tool to improve a meager programming environment, it has become one of the dominant languages of today. This paper studies its evolution.
Graphical models, exponential families, and variational inference
, 2008
"... The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building largescale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fiel ..."
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Cited by 800 (26 self)
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The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building largescale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fields, including bioinformatics, communication theory, statistical physics, combinatorial optimization, signal and image processing, information retrieval and statistical machine learning. Many problems that arise in specific instances — including the key problems of computing marginals and modes of probability distributions — are best studied in the general setting. Working with exponential family representations, and exploiting the conjugate duality between the cumulant function and the entropy for exponential families, we develop general variational representations of the problems of computing likelihoods, marginal probabilities and most probable configurations. We describe how a wide varietyof algorithms — among them sumproduct, cluster variational methods, expectationpropagation, mean field methods, maxproduct and linear programming relaxation, as well as conic programming relaxations — can all be understood in terms of exact or approximate forms of these variational representations. The variational approach provides a complementary alternative to Markov chain Monte Carlo as a general source of approximation methods for inference in largescale statistical models.
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