Second-order and Fluctuation-induced First-order Phase Transitions with Functional Renormalization Group Equations

We investigate phase transitions in scalar field theories using the functional renormalization group (RG) equation. We analyze a system with U(2)xU(2) symmetry, in which there is a parameter $\lambda_2$ that controls the strength of the first-order phase transition driven by fluctuations. In the limit of \lambda_2\to0$, the U(2)xU(2) theory is reduced to an O(8) scalar theory that exhibits a second-order phase transition in three dimensions. We develop a new insight for the understanding of the fluctuation-induced first-order phase transition as a smooth continuation from the standard RG flow in the O(8) system. In our view from the RG flow diagram on coupling parameter space, the region that favors the first-order transition emerges from the unphysical region to the physical one as \lambda_2 increases from zero. We give this interpretation based on the Taylor expansion of the functional RG equations up to the fourth order in terms of the field, which encompasses the$\epsilon$-expansion results. We compare results from the expansion and from the full numerical calculation and find that the fourth-order expansion is only of qualitative use and that the sixth-order expansion improves the quantitative agreement.Comment: 15 pages, 10 figures, major revision; discussions on O(N) models reduced, a summary section added after Introduction, references added; to appear in PR

Belief Revision, Minimal Change and Relaxation: A General Framework based on Satisfaction Systems, and Applications to Description Logics

Belief revision of knowledge bases represented by a set of sentences in a given logic has been extensively studied but for specific logics, mainly propositional, and also recently Horn and description logics. Here, we propose to generalize this operation from a model-theoretic point of view, by defining revision in an abstract model theory known under the name of satisfaction systems. In this framework, we generalize to any satisfaction systems the characterization of the well known AGM postulates given by Katsuno and Mendelzon for propositional logic in terms of minimal change among interpretations. Moreover, we study how to define revision, satisfying the AGM postulates, from relaxation notions that have been first introduced in description logics to define dissimilarity measures between concepts, and the consequence of which is to relax the set of models of the old belief until it becomes consistent with the new pieces of knowledge. We show how the proposed general framework can be instantiated in different logics such as propositional, first-order, description and Horn logics. In particular for description logics, we introduce several concrete relaxation operators tailored for the description logic \ALC{} and its fragments \EL{} and \ELext{}, discuss their properties and provide some illustrative examples

A revision of the Generalized Uncertainty Principle

The Generalized Uncertainty Principle arises from the Heisenberg Uncertainty Principle when gravity is taken into account, so the leading order correction to the standard formula is expected to be proportional to the gravitational constant $G_N = L_{Pl}^2$. On the other hand, the emerging picture suggests a set of departures from the standard theory which demand a revision of all the arguments used to deduce heuristically the new rule. In particular, one can now argue that the leading order correction to the Heisenberg Uncertainty Principle is proportional to the first power of the Planck length $L_{Pl}$. If so, the departures from ordinary quantum mechanics would be much less suppressed than what is commonly thought.Comment: 6 pages, 1 figur

Hydrostatic figure of the earth: Theory and results

The complete development of the mathematical theory of hydrostatic equilibrium for the earth is recounted. Modifications of the first order theory are given along with the subsequent extension to the second order. In addition, the equations are presented which resulted from a revision of the second order theory to suit the new applications and data types of the post-artificial earth satellite era

Tractability of Theory Patching

In this paper we consider the problem of `theory patching', in which we are given a domain theory, some of whose components are indicated to be possibly flawed, and a set of labeled training examples for the domain concept. The theory patching problem is to revise only the indicated components of the theory, such that the resulting theory correctly classifies all the training examples. Theory patching is thus a type of theory revision in which revisions are made to individual components of the theory. Our concern in this paper is to determine for which classes of logical domain theories the theory patching problem is tractable. We consider both propositional and first-order domain theories, and show that the theory patching problem is equivalent to that of determining what information contained in a theory is `stable' regardless of what revisions might be performed to the theory. We show that determining stability is tractable if the input theory satisfies two conditions: that revisions to each theory component have monotonic effects on the classification of examples, and that theory components act independently in the classification of examples in the theory. We also show how the concepts introduced can be used to determine the soundness and completeness of particular theory patching algorithms.Comment: See http://www.jair.org/ for any accompanying file

Nonleptonic B decays into two light mesons in soft-collinear effective theory

We consider nonleptonic B decays into two light mesons at leading order in soft-collinear effective theory, and show that the decay amplitudes are factorized to all orders in alpha_s. The operators for nonleptonic B decays in the full theory are first matched to the operators in SCET_I, which is the effective theory appropriate for sqrt{m_b Lambda} <mu <m_b with Lambda~0.5 GeV. We evolve the operators and the relevant time-ordered products in SCET_I to SCET_II, which is appropriate for mu < sqrt{m_b Lambda}. Using the gauge-invariant operators in SCET_II, we compute nonleptonic B decays in SCET, including the nonfactorizable spectator contributions and spectator contributions to the heavy-to-light form factor. As an application, we present the decay amplitudes for B ->pi,pi in soft-collinear effective theory.Comment: 42 pages, 5 figures, 2 tables Major revision of the manuscript. The idea of using SCET_I, and SCET_II is clearly presented. Some of the calculational steps are explicitly show