On Convergence Properties of Shannon Entropy

Convergence properties of Shannon Entropy are studied. In the differential setting, it is shown that weak convergence of probability measures, or convergence in distribution, is not enough for convergence of the associated differential entropies. A general result for the desired differential entropy convergence is provided, taking into account both compactly and uncompactly supported densities. Convergence of differential entropy is also characterized in terms of the Kullback-Liebler discriminant for densities with fairly general supports, and it is shown that convergence in variation of probability measures guarantees such convergence under an appropriate boundedness condition on the densities involved. Results for the discrete setting are also provided, allowing for infinitely supported probability measures, by taking advantage of the equivalence between weak convergence and convergence in variation in this setting.Comment: Submitted to IEEE Transactions on Information Theor

MDL Convergence Speed for Bernoulli Sequences

The Minimum Description Length principle for online sequence estimation/prediction in a proper learning setup is studied. If the underlying model class is discrete, then the total expected square loss is a particularly interesting performance measure: (a) this quantity is finitely bounded, implying convergence with probability one, and (b) it additionally specifies the convergence speed. For MDL, in general one can only have loss bounds which are finite but exponentially larger than those for Bayes mixtures. We show that this is even the case if the model class contains only Bernoulli distributions. We derive a new upper bound on the prediction error for countable Bernoulli classes. This implies a small bound (comparable to the one for Bayes mixtures) for certain important model classes. We discuss the application to Machine Learning tasks such as classification and hypothesis testing, and generalization to countable classes of i.i.d. models.Comment: 28 page

Circular dichroism of cholesteric polymers and the orbital angular momentum of light

We explore experimentally if the light's orbital angular momentum (OAM) interacts with chiral nematic polymer films. Specifically, we measure the circular dichroism of such a material using light beams with different OAM. We investigate the case of strongly focussed, non-paraxial light beams, where the spatial and polarization degrees of freedom are coupled. Within the experimental accuracy, we cannot find any influence of the OAM on the circular dichroism of the cholesteric polymer.Comment: 3 pages, 4 figure

Relating Agulhas leakage to the Agulhas Current retroflection location

The relation between the Agulhas Current retroflection location and the magnitude of Agulhas leakage, the transport of water from the Indian to the Atlantic Ocean, is investigated in a high-resolution numerical ocean model. Sudden eastward retreats of the Agulhas Current retroflection loop are linearly related to the shedding of Agulhas rings, where larger retreats generate larger rings. Using numerical Lagrangian floats a 37 year time series of the magnitude of Agulhas leakage in the model is constructed. The time series exhibits large amounts of variability, both on weekly and annual time scales. A linear relation is found between the magnitude of Agulhas leakage and the location of the Agulhas Current retroflection, both binned to three month averages. In the relation, a more westward location of the Agulhas Current retroflection corresponds to an increased transport from the Indian Ocean to the Atlantic Ocean. When this relation is used in a linear regression and applied to almost 20 years of altimetry data, it yields a best estimate of the mean magnitude of Agulhas leakage of 13.2 Sv. The early retroflection of 2000, when Agulhas leakage was probably halved, can be identified using the regression

The effect of pitched and vertical ladder ergometer climbing on cardiorespiratory and psychophysical variables.

This study aimed to assess whether modifying the pitch of a 75° ladder ergometer to vertical had a cardiorespiratory or psychophysical effect on climbing. Nine male participants climbed a ladder ergometer at 75° and subsequently at 90°, adjusted for an equivalent vertical climb rate, completing three climbing bouts at different vertical speeds. One participant dropped out being unable to complete the climb under the 90° condition. Each was monitored for heart rate (HR), V˙O2 and rating of perceived exertion (RPE). Results showed vertical climbing induced higher V˙O2 (mean increase 17.3%), higher HR (mean increase 15.8%), and higher RPE at all speeds and that moving from 75° to vertical exacerbates the effect of speed on the cardiorespiratory response to climbing. This may be explained by increased force production required to maintain balance in a vertical climbing position when the body's centre of mass is not above the feet

C2C_2-cofiniteness of 2-cyclic permutation orbifold models

In this article, we consider permutation orbifold models of $C_2$-cofinite vertex operator algebras of CFT type. We show the $C_2$-cofiniteness of the 2-cyclic permutation orbifold model $(V\otimes V)^{S_2}$ for an arbitrary $C_2$-cofinite simple vertex operator algebra $V$ of CFT type. We also give a proof of the $C_2$-cofiniteness of a $\Z_2$-orbifold model $V_L^+$ of the lattice vertex operator algebra $V_L$ associated with a rank one positive definite even lattice $L$ by using our result and the $C_2$-cofiniteness of $V_L$.Comment: 25 pages, no figure, some typo are correcte