On the Behavior of the Generalized Alignment Index (GALI) Method for Regular Motion in Multidimensional Hamiltonian Systems

We investigate the behavior of the Generalized Alignment Index of order k (GALIk ) for regular orbits of multidimensional Hamiltonian systems. The GALIk is an efficient chaos indicator, which asymptotically attains positive values for regular motion when 2≤k ≤N, with N being the dimension (D) of the torus on which the motion occurs. By considering several regular orbits in the neighborhood of two typical simple, stable periodic orbits of the Fermi-Pasta-Ulam-Tsingou (FPUT) β model for various values of the system's degrees of freedom, we show that the asymptotic GALIk values decrease when the order k of the index increases and when the orbit's energy approaches the periodic orbit's destabilization energy where the stability island vanishes, while they increase when the considered regular orbit moves further away from the periodic one for a fixed energy. In addition, by performing extensive numerical simulations we show that the behavior of the index does not depend on the choice of the initial deviation vectors needed for its evaluation.

Contextualism, epistemic, recent work on

What you know at a given time depends of course on features of your context. You can’t know you see a fire, for example, unless there is a fire there, which involves how it is in your context. Everyone would surely agree on that much. Interesting disagreement sets in on the ways in which your knowledge, and even your justification, can depend on features of your context. Some hold that, beyond the obvious dependence of knowledge on truth (often enough external truth), there is also the dependence of knowledge on external causal or counterfactual properties of the constitutive belief. Thus, whether S knows that p at t, Kp, has been said to depend on one or more of the following: - whether Bp (S’s belief that p) is accidentally true; - whether Bp is caused by the fact that p; - whether Bp tracks the truth; - whether Bp derives from a reliable process. And these are all often said to be matters of the (external) context of the believer at the time of the belief. Internalists have reached near-consensus in conceding that externalism is (or may be) right about knowledge, while retreating to internalism of justification. More specifically, it is conceded that the total epistemic condition that a belief must satisfy in order to qualify as knowledge is (or may be) often enough an external condition that goes beyond the external truth of that belief. According to internalists it is whether one is epistemically justified that depends only on what is internal to one’s mind. Externalists disagree. In their view even what one is justified in believing depends on features of one’s (external) context. Such externalists thus accept a kind of ‘contextualism’ that concerns the dependence of the epistemic status of beliefs on the (external) context of the believer. Externalism with regard to a given epistemic status (knowledge, for example, or justification) holds that it does depend on the external context of the believer, while internalism holds that it does not. The view that a given epistemic status depends thus on context is generally known as ‘externalism’, however, and not as ‘contextualism’. The latter term is reserved mostly for a different view, one about the truth conditions of epistemic attributions. According to such attributor-contextualism, the truth of the attribution of a certain epistemic status by the use of a certain epistemic term (say, ‘knows’) depends on features of the context of attribution that involve not just the truth, or the content, of the subject’s belief; indeed the content (in one sense) is itself determined by such contextual features. The main terms appealed to by various forms of externalism do in fact seem thus attributor-contextual. This is quite plausibly true of ‘x is accidental’, of ‘x is caused by y’, and of ‘RP is a reliable process’. And it is even arguably true of ‘x tracks y’, that is of ‘x would be so if y were so, and would not be so if y were not so’. And it seems no less true of the most plausible forms of traditional internalism of knowledge, with their dependence on a ‘sufficient justification’ requirement. Thus, to consider just one example, the truth of an utterance of ‘x is accidental’, in at least one sense of that expression, plausibly depends on background knowledge possessed by the attributor (or perhaps his audience, or his relevant community). Take the utterance about a meeting by Sally and Tom at the supermarket that this was ‘accidental’. This may be true in the mouth of either Sally or Tom, each of whom is ignorant of the regular orbit of the other, but it may well be false if affirmed by an agent of the detective agency following them both. Nevertheless, we can still distinguish externalism from contextualism. Any form of externalism is still a view in epistemology proper, one that lays down externalist conditions, necessary or sufficient, for a given epistemic status, and does so in the object language. So, it may say that a belief amounts to knowledge only if it derives from a reliable source. Contextualism, by contrast, ascends to the metalanguage, where one speaks about the object language, saying perhaps that a speaker’s utterance of the form ‘S knows that p’ is true if the subject’s belief is related in a certain way to the speaker; thus, it might be required that the subject’s belief have a source whose reliability lies above a threshold set by the speaker’s community. But the requirement of a reliable source is just one example of a possible externalist condition. Alternatively, the externalist may lay down the condition that the belief not be accidentally true, for example, or that it be caused by the fact that constitutes its content, or that it tracks the truth. Of course, in specifying its favoured condition – say ‘reliable derivation’ – it depends on the context of use to provide an index only relative to which does the claim have a truth-value. Even so, we can still distinguish the question expressed in a certain context by a contextual expression – that is one posed in the object language – from metalinguistic questions as to the contexts in which that expression can be used for what semantical effects.

REGULAR ORBITS OF COPRIME LINEAR GROUPS IN LARGE CHARACTERISTIC

We prove that a finite coprime linear group $G$ in characteristic $p\geq \frac{1}{2}(|G|-1)$ has a regular orbit. This bound on $p$ is best possible. We also give an application to blocks with abelian defect groups.

Traces for Star Products on the Dual of a Lie Algebra

In this paper, we describe all traces for the BCH star-product on the dual of a Lie algebra. First we show by an elementary argument that the BCH as well as the Kontsevich star-product are strongly closed if and only if the Lie algebra is unimodular. In a next step we show that the traces of the BCH star-product are given by the ad-invariant functionals. Particular examples are the integration over coadjoint orbits. We show that for a compact Lie group and a regular orbit one can even achieve that this integration becomes a positive trace functional. In this case we explicitly describe the corresponding GNS representation. Finally we discuss how invariant deformations on a group can be used to induce deformations of spaces where the group acts on.