In this paper, by using Zalcman Lemma, we obtain some normal criterions of meromorphic functions concerning shared fixed-points, which improves some earlier related results.
In this paper, by using Zalcman Lemma, we obtain some normal criterions of meromorphic functions concerning shared fixed-points, which improves some earlier related results.
Bell non-local correlations cannot be naturally explained in a fixed causal structure. This serves as a motivation for considering models where no global assumption is made beyond logical consistency. The assumption o...
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Bell non-local correlations cannot be naturally explained in a fixed causal structure. This serves as a motivation for considering models where no global assumption is made beyond logical consistency. The assumption of a fixed causal order between a set of parties, together with free randomness, implies device-independent inequalities-just as the assumption of locality does. It is known that local validity of quantum theory is consistent with violating such inequalities. Moreover, for three parties or more, even the (stronger) assumption of local classical probability theory plus logical consistency allows for violating causal inequalities. Here, we show that a classical environment (with which the parties interact), possibly containing loops, is logically consistent if and only if whatever the involved parties do, there is exactly one fixed-point, the latter being representable as a mixture of deterministic fixed-points. We further show that the non-causal view allows for a model of computation strictly more powerful than computation in a world of fixed causal orders.
We prove that, if Gamma is a finite connected 3-valent vertex-transitive, or 4-valent vertex-and edge-transitive graph, then either Gamma is part of a well-understood family of graphs, or every non-identity automorphi...
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We prove that, if Gamma is a finite connected 3-valent vertex-transitive, or 4-valent vertex-and edge-transitive graph, then either Gamma is part of a well-understood family of graphs, or every non-identity automorphism of Gamma fixes at most 1/3 of the edges. This answers a question proposed by Primoz Potocnik and the third author.
This paper examines the dynamical systems of the Sigmoid Beverton-Holt model with overlapping generations in the projective line P-1(Q(p)). In particular, an in-depth analysis of the corresponding fixed-points and the...
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This paper examines the dynamical systems of the Sigmoid Beverton-Holt model with overlapping generations in the projective line P-1(Q(p)). In particular, an in-depth analysis of the corresponding fixed-points and the dynamics surrounding them is carried out. An extensive array of illustrative examples will also be discussed.
In this paper, we study the normality criteria of meromorphic functions concerning shared fixed-points, we obtain: Let F be a family of meromorphic functions defined in a domain D. Let n, k ≥ 2 be two positive intege...
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In this paper, we study the normality criteria of meromorphic functions concerning shared fixed-points, we obtain: Let F be a family of meromorphic functions defined in a domain D. Let n, k ≥ 2 be two positive integers. For every f ∈ F, all of whose zeros have multiplicity at least (nk+2)/(n-1). If f(f(k))nand g(g(k))nshare z in D for each pair of functions f and g, then F is normal.
The synthesis of maximally-permissive controllers in infinite-state systems has many practical applications. Such controllers directly correspond to maximal winning strategies in logically specified infinite-state two...
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ISBN:
(纸本)9781450385626
The synthesis of maximally-permissive controllers in infinite-state systems has many practical applications. Such controllers directly correspond to maximal winning strategies in logically specified infinite-state two-player games. In this paper, we introduce a tool called GenSys which is a fixed-point engine for computing maximal winning strategies for players in infinite-state safety games. A key feature of GenSys is that it leverages the capabilities of existing off-the-shelf solvers to implement its fixed point engine. GenSys outperforms state-of-the-art tools in this space by a significant margin. Our tool has solved some of the challenging problems in this space, is scalable, and also synthesizes compact controllers. These controllers are comparatively small in size and easier to comprehend. GenSys is freely available for use and is available under an open-source license.
In this paper, we develop an Isabelle/HOL library of order-theoretic fixedpoint theorems. We keep our formalization as general as possible: we reprove several well-known results about complete orders, often with only ...
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In this paper, we develop an Isabelle/HOL library of order-theoretic fixedpoint theorems. We keep our formalization as general as possible: we reprove several well-known results about complete orders, often with only antisymmetry or attractivity, a mild condition implied by either antisymmetry or transitivity. In particular, we generalize various theorems ensuring the existence of a quasi-fixed point of monotone maps over complete relations, and show that the set of (quasi-)fixedpoints is itself complete. This result generalizes and strengthens theorems of Knaster-Tarski, Bourbaki-Witt, Kleene, Markowsky, Pataraia, Mashburn, Bhatta-George, and Stouti-Maaden.
We prove a topological two-way characterization of the existence of fixed-points, without using linear or convexity structures and provide applications in optimization-related problems. Such a characterization is also...
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We prove a topological two-way characterization of the existence of fixed-points, without using linear or convexity structures and provide applications in optimization-related problems. Such a characterization is also demonstrated for a fixed-component point, a slight generalization of a fixed point. (C) 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
It follows from known results in the literature that least and greatest fixed-points of monotone polynomials on Heyting algebras-that is, the algebraic models of the Intuitionistic Propositional Calculus-always exist,...
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It follows from known results in the literature that least and greatest fixed-points of monotone polynomials on Heyting algebras-that is, the algebraic models of the Intuitionistic Propositional Calculus-always exist, even when these algebras are not complete as lattices. The reason is that these extremal fixed-points are definable by formulas of the IPC. Consequently, the mu-calculus based on intuitionistic logic is trivial, every mu-formula being equivalent to a fixed-point free formula. In the first part of this article, we give an axiomatization of least and greatest fixed-points of formulas, and an algorithm to compute a fixed-point free formula equivalent to a given mu-formula. The axiomatization of the greatest fixed-point is simple. The axiomatization of the least fixed-point is more complex, in particular every monotone formula converges to its least fixed-point by Kleene's iteration in a finite number of steps, but there is no uniform upper bound on the number of iterations. The axiomatization yields a decision procedure for the mu-calculus based on propositional intuitionistic logic. The second part of the article deals with closure ordinals of monotone polynomials on Heyting algebras and of intuitionistic monotone formulas;these are the least numbers of iterations needed for a polynomial/formula to converge to its least fixed-point. Mirroring the elimination procedure, we show how to compute upper bounds for closure ordinals of arbitrary intuitionistic formulas. For some classes of formulas, we provide tighter upper bounds that, in some cases, we prove exact.
An expression such as for all x(P(x) phi(P)), where P occurs in 0(P), does not always define P. When such an expression implicitly defines P. in the sense of Beth (1953)[1] and Padoa (1900) [13], we call it a recursi...
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An expression such as for all x(P(x) <-> phi(P)), where P occurs in 0(P), does not always define P. When such an expression implicitly defines P. in the sense of Beth (1953)[1] and Padoa (1900) [13], we call it a recursive definition. In the Least fixed-Point Logic (LFP), we have theories where interesting relations can be recursively defined (Ebbinghaus, 1995 [4], Libkin, 2004 [12]). We will show that for some sorts of recursive definitions there are explicit definitions on sufficiently strong theories of LFP. It is known that LFP, restricted to finite models, does not have Beth's Definability Theorem (Gurevich, 1996[7], Hodkinson, 1993[8], Dawar, 1995[3]). Beth's Definability Theorem states that, if a relation is implicitly defined, then there is an explicit definition for it. We will also give a proof that Beth's Definability Theorem fails for LFP without this finite model restriction. We will investigate fragments of LFP for which Beth's Definability Theorem holds, specifically theories whose models are well-founded structures. (C) 2011 Elsevier B.V. All rights reserved.
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