Normal Subgroups of B u Aut(Ω)

Logical approaches-and ontologies in particular-offer a well-adapted framework for representing knowledge present on the Semantic Web ( Open image in new window ). These ontologies are formulated in Open image in new window ( Open image in new window ), which are based on expressive Open image in new window ( Open image in new window ). Open image in new window are a subset of Open image in new window ( Open image in new window ) that provides decidable reasoning. Based on Open image in new window , it is possible to rely on inference mechanisms to obtain new knowledge from axioms, rules and facts specified in the ontologies. However, these classical inference mechanisms do not deal with : Open image in new window probabilities. Several works recently targeted those issues (i.e. Open image in new window , Open image in new window , Open image in new window , etc.), but none of them combines Open image in new window with Open image in new window ( Open image in new window ) formalism. Several open source software packages for Open image in new window are available (e.g. Open image in new window , Open image in new window , Open image in new window , etc.). In this paper, we present Open image in new window , a Java framework for reasoning with probabilistic information in the Open image in new window . Open image in new window incorporate three open source software packages for Open image in new window , which is able to reason with uncertainty information, showing that it can be used in several real-world domains. We also show several experiments of our tool with different ontologies.

In this paper, we give a global view of the results we have obtained in relation with a remarkable class of submartingales, called (Σ), and which are stated in (Najnudel and Nikeghbali, 0906.1782 (2009), 0910.4959 (2009), 0911.2571 (2009) and 0911.4365 (2009)). More precisely, we associate to a given submartingale in this class (Σ), defined on a filtered probability space Open image in new window , satisfying some technical conditions, a σ-finite measure Open image in new window on Open image in new window , such that for all t≥0, and for all events Open image in new window : Open image in new window where g is the last hitting time of zero of the process X. This measure Open image in new window has already been defined in several particular cases, some of them are involved in the study of Brownian penalisation, and others are related with problems in mathematical finance. More precisely, the existence of Open image in new window in the general case solves a problem stated by D. Madan, B. Roynette and M. Yor, in a paper studying the link between Black-Scholes formula and last passage times of certain submartingales. Once the measure Open image in new window is constructed, we define a family of nonnegative martingales, corresponding to the local densities (with respect to ℙ) of the finite measures which are absolutely continuous with respect to Open image in new window . We study in detail the relation between Open image in new window and this class of martingales, and we deduce a decomposition of any nonnegative martingale into three parts, corresponding to the decomposition of finite measures on Open image in new window as the sum of three measures, such that the first one is absolutely continuous with respect to ℙ, the second one is absolutely continuous with respect to Open image in new window and the third one is singular with respect to ℙ and Open image in new window . This decomposition can be generalized to supermartingales. Moreover, if under ℙ, the process (X t ) t≥0 is a diffusion satisfying some technical conditions, one can state a penalisation result involving the measure Open image in new window , and generalizing a theorem given in (Najnudel et al., A Global View of Brownian Penalisations, 2009). Now, in the construction of the measure Open image in new window , we encounter the following problem: if Open image in new window is a filtered probability space satisfying the usual assumptions, then it is usually not possible to extend to Open image in new window (the σ-algebra generated by Open image in new window ) a coherent family of probability measures (ℚ t ) indexed by t≥0, each of them being defined on Open image in new window . That is why we must not assume the usual assumptions in our case. On the other hand, the usual assumptions are crucial in order to obtain the existence of regular versions of paths (typically adapted and continuous or adapted and cadlag versions) for most stochastic processes of interest, such as the local time of the standard Brownian motion, stochastic integrals, etc. In order to fix this problem, we introduce another augmentation of filtrations, intermediate between the right continuity and the usual conditions, and call it N-augmentation in this paper. This augmentation has also been considered by Bichteler (Stochastic integration and stochastic differential equations, 2002). Most of the important results of the theory of stochastic processes which are generally proved under the usual augmentation still hold under the N-augmentation; moreover this new augmentation allows the extension of a coherent family of probability measures whenever this is possible with the original filtration.

Let {\mathcal L}^p_{\alpha}(U^n) denote the class of all measurable functions defined on the unit polydisc U^n=\{z\in {\bf C}^n\, \big| \;|z_i|<1,\ i=1,...,n\} such that \|f\|^p_{{\mathcal L}_{\alpha}(U^n)}=\int_{U^n}|f(z)|^p\prod_{j=1}^n (1-|z_j|^2)^{\alpha_j}dm(z_j)<\infty, where \alpha_j>-1 , j=1,...,n , and dm(z_j) is the normalized area measure on the unit disk U , H(U^n) the class of all holomorphic functions on U^n , and let {\mathcal A}^p_{\alpha}(U^n)={\mathcal L}^p_{\alpha}(U^n) \cap H(U^n) (the weighted Bergman space). In this paper we prove that for p\in (0,\infty), f\in {\mathcal A}^p_{\alpha}(U^n) if and only if the functions \prod_{j\in S}(1-|z_j|^2)\frac{\partial ^{|S|} f} {\prod_{j\in S}\partial z_j}\big(\chi_S(1)z_1, \chi_S(2)z_2,..., \chi_S(n)z_n\big) belong to the space {\cal L}^p_{\alpha}(U^n) for every S\subseteq \{1,2,...,n\}, where \chi_S(\cdot) is the characteristic function of S, |S| is the cardinal number of S, and \prod_{j\in S}\partial z_j=\partial z_{j_1}\cdots\partial z_{j_{|S|}}, where j_k\in S, \, k=1,...,|S|. This result extends Theorem 22 of Kehe Zhu in Trans. Amer. Math. Soc. 309 (1988) (1), 253–268, when p\in (0,1). Also in the case p\in [1,\infty) , we present a new proof.

Let Open image in new window denote the set of Liouville numbers. For a dimension function h, we write Open image in new window for the h-dimensional Hausdorff measure of Open image in new window. In previous work, the exact ``cut-point'' at which the Hausdorff measure Open image in new window of Open image in new window drops from infinity to zero has been located for various classes of dimension functions h satisfying certain rather restrictive growth conditions. In the paper, we locate the exact ``cut-point'' at which the Hausdorff measure Open image in new window of Open image in new window drops from infinity to zero for all dimension functions h. Namely, if h is a dimension function for which the function Open image in new window increases faster than any power function near 0, then Open image in new window, and if h is a dimension function for which the function Open image in new window increases slower than some power function near 0, then Open image in new window. This provides a complete characterization of all Hausdorff measures Open image in new window of Open image in new window without assuming anything about the dimension function h, and answers a question asked by R. D. Mauldin. We also show that if Open image in new window then Open image in new window does not have σ-finite Open image in new window measure. This answers another question asked by R. D. Mauldin.

Let X be any Banach space and T a bounded operator on X. An extensionOpen image in new window of the pair (X,T) consists of a Banach space Open image in new window in which X embeds isometrically through an isometry i and a bounded operatorOpen image in new window on Open image in new window such that Open image in new window When X is separable, it is additionally required that Open image in new window be separable. We say that Open image in new window is a topologically transitive extension of (X, T) when Open image in new window is topologically transitive on Open image in new window, i.e. for every pair Open image in new window of non-empty open subsets of Open image in new window there exists an integer n such that Open image in new window is non-empty. We show that any such pair (X,T) admits a topologically transitive extension Open image in new window, and that when H is a Hilbert space, (H,T) admits a topologically transitive extension Open image in new window where Open image in new window is also a Hilbert space. We show that these extensions are indeed chaotic.

Turing Tumble is a toy gravity-fed mechanical computer (similar to the classic Open image in new window , but including additional types of pieces such as gears), in which marbles roll down a board, along paths determined by the locations of ramps, toggles and gears, which are placed by the “programmer,” and by their current states, which are altered by the passing marbles. Aaronson proved that a Open image in new window decision problem (viz., will any marbles reach the sink?) is Open image in new window -Complete, i.e., equivalent to evaluating comparator circuits, and posed the question of what additional functionality would raise the machine’s computational power beyond Open image in new window , speculating that a capability for toggles to affect one another’s states (which Open image in new window gears happen to provide) might suffice. This turns out to be so: we show, though a simple reduction from a variant of the circuit value problem ( Open image in new window ), that the Open image in new window decision problem is Open image in new window -Complete. The two models also differ in complexity when exponentially (or unboundedly) many marbles are permitted: while Open image in new window remains in Open image in new window , Open image in new window becomes Open image in new window -Complete.

We investigate experimentally the effect of aspect ratio ( Open image in new window ) on the time-varying, three-dimensional flow structure of flat-plate wings rotating from rest at 45° angle of attack. Plates of Open image in new window = 2 and 4 are tested in a 50 % by mass glycerin–water mixture, with a total rotation of ϕ = 120° and a matched tip Reynolds number of 5,000. The time-varying, three-component volumetric velocity field is reconstructed using phase-locked, phase-averaged stereoscopic digital particle image velocimetry in multiple, closely-spaced chordwise planes. The vortex structure is analyzed using the \(\mathcal{Q}\)-criterion, helicity density, and spanwise quantities. For both Open image in new window s, the flow initially consists of a connected and coherent leading-edge vortex (LEV), tip vortex (TV), and trailing-edge vortex (TEV) loop; the LEV increases in size with span and tilts aft. Smaller, discrete vortices are present in the separated shear layers at the trailing and tip edges, which wrap around the primary TEV and TV. After about ϕ = 20°, the outboard-span LEV lifts off the plate and becomes arch-like. A second, smaller LEV and the formation of corner vortex structures follow. For Open image in new window = 4, the outboard LEV moves farther aft, multiple LEVs form ahead of it, and after about ϕ = 50° a breakdown of the lifted-off LEV and the TV occurs. However, for Open image in new window = 2, the outboard LEV lift-off is not progressive, and the overall LEV-TV flow remains more coherent and closer to the plate, with evidence of breakdown late in the motion. Inboard of about 50 % span, the Open image in new window = 4 LEV is stable for the motion duration. Up to approximately 60 % span, the Open image in new window = 2 LEV is distinct from the TV and is similarly stable. The Open image in new window = 2 LEV exhibits substantially higher spanwise vorticity and velocity. The latter possesses a “four-lobed” distribution at the periphery of the LEV core having adjacent positive (outboard) and negative (inboard) components, corresponding to a helical streamline structure. Both Open image in new window s show substantial root-to-tip velocity aft of the stable LEV, which drives outboard spanwise vorticity flux; flux toward the root is also present in the front portion of the LEV. For Open image in new window = 2, there is a strong flux of spanwise vorticity from the outboard LEV to the tip, which may mitigate LEV lift-off and is not found for Open image in new window = 4. The TV circulation for each Open image in new window is similar in magnitude and growth when plotted versus the chord lengths travelled by the tip, prior to breakdown. Streamwise vorticity due to the TV induces high spanwise velocity, and for Open image in new window = 2, the tilted LEV creates further streamwise vorticity which corresponds well to spanwise-elongated regions of spanwise velocity. For Open image in new window = 2, the TV influences a relatively greater portion of the span and is more coherent at later times, which coupled with the tilted LEV strongly contributes to the higher overall spanwise velocity and vorticity flux.

Let Open image in new window be functors. We know that the class [F, G] of all functorial morphisms from F into G is not always a set, so that we can not speak in general about the category of all functors from Open image in new window into Open image in new window et. But instead of all functors from Open image in new window into Open image in new window we can consider only so-called proper functors, which were defined by Isbell [3]. In this way we can define a category Open image in new window , the category of all proper functors from Open image in new window into Open image in new window . This category contains Open image in new window as a full subcategory and it coincides with the category Open image in new window in the case when Open image in new window is a small category.

In this paper we extend results from Semigroup Theory on existence and characterization of attractors in order to include multivalued semigroups T(t) defined by generalized semiflows Open image in new window. In particular we show that, if Open image in new window is continuous, possesses a Lyapunov function, and Open image in new window has a global attractor Open image in new window which is maximal compact invariant, then Open image in new window = Wu(Z(Open image in new window)), where Z(Open image in new window) is the stationary solutions set and Wu(Z(Open image in new window)) is the unstable set of Z(Open image in new window). We introduce the Open image in new window-attractor concept which does not enjoy any uniformity on time of attraction and we prove, under suitable conditions, that the global Open image in new window-attractor \(\widehat{N}\) is the set of asymptotic states described by Z(Open image in new window).

Let Open image in new window be a commutative subspace lattice generated by finite many commuting independent nests on a complex separable Hilbert space Open image in new window with Open image in new window, and Open image in new window the associated CSL algebra. It is proved that every Lie triple derivation from Open image in new window into any σ-weakly closed algebra Open image in new window containing Open image in new window is of the form X→XT−TX+h(X)I, where Open image in new window and h is a linear mapping from Open image in new window into ℂ such that h([[A,B],C])=0 for all Open image in new window.

This paper helps to bridge the gap between (i) the use of logic for specifying program semantics and performing program analysis, and (ii) abstract interpretation. Many operations needed by an abstract interpreter can be reduced to the problem of symbolic abstraction: the symbolic abstraction of a formula ϕ in logic Open image in new window , denoted by Open image in new window , is the most-precise value in abstract domain Open image in new window that over-approximates the meaning of ϕ. We present a parametric framework that, given Open image in new window and Open image in new window , implements Open image in new window . The algorithm computes successively better over-approximations of Open image in new window . Because it approaches Open image in new window from “above”, if it is taking too much time, a safe answer can be returned at any stage.Moreover, the framework is“dual-use”: in addition to its applications in abstract interpretation, it provides a new way for an SMT (Satisfiability Modulo Theories) solver to perform unsatisfiability checking: given Open image in new window , the condition Open image in new window implies that ϕ is unsatisfiable.

Given H:ℝ3→ℝ of class C1 and bounded, we consider a sequence (un) of solutions of the H-system Open image in new window in the unit open disc Open image in new window satisfying the boundary condition un=γn on ∂Open image in new window. In the first part of this paper, assuming that (un) is bounded in H1(Open image in new window,ℝ3) we study the behavior of (un) when the boundary data γn shrink to zero. We show that either un→0 strongly in H1(Open image in new window,ℝ3) or un blows up at least one H-bubble ω, namely a nonconstant, conformal solution of the H-system on ℝ2. Under additional assumptions on H, we can obtain more precise information on the blow up. In the second part of this paper we investigate the multiplicity of solutions for the Dirichlet problem on the disc with small boundary datum. We detect a family of nonconstant functions H (even close to a nonzero constant in any reasonable topology) for which the Dirichlet problem cannot admit a ``large'' solution at a mountain pass level when the boundary datum is small.

We introduce in a natural way the notion of measure with bounded variation with respect to a normed ideal of operators and prove that for each maximal normed ideal of operators ( Open image in new window , Open image in new window ), is true the following result: If U ∈ L(C(T,X), Y) with G the representing measure of U and G : Σ → ( Open image in new window (X, Y), Open image in new window ) has bounded variation, then U ∈ Open image in new window (C(T,X), Y). As an application of this result we prove that an injective tensor product of an integral operator with an operator belonging to a maximal normed ideal of operators ( Open image in new window , Open image in new window ) belongs also to ( Open image in new window , Open image in new window ).

Let Φ be a system of ideals in a commutative Noetherian ring R, and let Open image in new window be a Serre subcategory of R-modules. We set $$ H_\Phi ^i ( \cdot , \cdot ) = \mathop {\lim }\limits_{\overrightarrow {\mathfrak{b} \in \Phi } } Ext_R^i (R/\mathfrak{b}| \otimes R \cdot , \cdot ). $$ . Suppose that a is an ideal of R, and M and N are two R-modules such that M is finitely generated and N ∈ Open image in new window. It is shown that if the functor \( D_\Phi ( \cdot ) = \mathop {\lim }\limits_{\overrightarrow {\mathfrak{b} \in \Phi } } Hom_R (\mathfrak{b}, \cdot ) \) is exact, then, for any \( \mathfrak{b} \in \Phi ,Ext_R^j (R/\mathfrak{b},H_\Phi ^i (M,N)) \) ∈ Open image in new window for all i, j ≥ 0. It is also proved that if there is a nonnegative integer t such that \( H_\mathfrak{a}^i (M,N) \) ∈ Open image in new window for all i < t, then \( Hom_R (R/\mathfrak{a},H_\mathfrak{a}^t (M,N)) \) ∈ Open image in new window, provided that Open image in new window is contained in the class of weakly LaskerianR-modules. Finally, it is shown that if L is an R-module and t is the infimum of the integers i such that \( H_\mathfrak{a}^i (L) \) ∈ Open image in new window, then \( Ext_R^j (R/\mathfrak{a},H_\mathfrak{a}^t (M,L)) \) ∈ Open image in new window if and only if \( Ext_R^j (R/\mathfrak{a},Hom_R (M,H_\mathfrak{a}^t (L))) \) ∈ Open image in new window for all j ≥ 0.

Let Open image in new window denote the set of normalized analytic functions f(z) = z + Σk=2∞akzk in the unit disk |z| < 1, and let sn(z) represent the nth partial sum of f(z). Our first objective of this note is to obtain a bound for \(|\frac{{s_n (z)}} {{f(z)}} - 1| \) when f ∈ Open image in new window is univalent in ⅅ. Let Open image in new window denote the set of all f ∈ Open image in new window in ⅅ satisfying the condition $$\left| {f'(z)\left( {\frac{z} {{f(z)}}} \right)^2 - 1} \right| < 1$$ for |z| 1. In case f″ (0) = 0, we find that all corresponding sections sn of f ∈ Open image in new window are in Open image in new window in the disk \(|z| 1/2 in the disk \( \left| z \right| < \sqrt {\sqrt 5 - 2}\). Finally, we establish a necessary coefficient condition for functions in Open image in new window and the related radius problem for an associated subclass of Open image in new window. In result, we see that if f ∈ Open image in new window then for n ≥ 3 we have $$\left| {\frac{{f(z)}} {{s_n (z)}} - \frac{4} {3}} \right| < \frac{2} {3}for|z| < r_n : = 1 - \frac{{2\log n}} {n}$$