A Note on Causation versus Correlation in an Extreme Situation

Recently, it has been shown that the information flow and causality between two time series can be inferred in a rigorous and quantitative sense, and, besides, the resulting causality can be normalized. A corollary that follows is, in the linear limit, causation implies correlation, while correlation does not imply causation. Now suppose there is an event A taking a harmonic form (sine/cosine), and it generates through some process another event B so that B always lags A by a phase of π/2. Here the causality is obviously seen, while by computation the correlation is, however, zero. This apparent contradiction is rooted in the fact that a harmonic system always leaves a single point on the Poincaré section; it does not add information. That is to say, though the absolute information flow from A to B is zero, i.e., TA→B=0, the total information increase of B is also zero, so the normalized TA→B, denoted as τA→B, takes the form of 00. By slightly perturbing the system with some noise, solving a stochastic differential equation, and letting the perturbation go to zero, it can be shown that τA→B approaches 100%, just as one would have expected.

Higher Integrability for Very Weak Solutions of InhomogeneousA-Harmonic Form Equations

The higher integrability for very weak solutions ofA-harmonic form equationsd*A(x,u,du)=B(x,u,du)has been proved.

Increased Pulmonary Venous Resistance in Morbidly Obese Patients without Daytime Hypoxia

Background The pulmonary artery (PA) diastolic-pulmonary capillary wedge pressure (PAD-PCWP) gradient has been shown to be increased in morbidly obese patients without daytime hypoxia. In sepsis, the increased pulmonary venous resistance (PvR) contributes to increases in PAD-PCWP gradient. In addition, the obesity-related endotoxemia is known to be involved in the pathophysiology of metabolic syndrome in obesity. Therefore, it is possible that the increased PvR contributes to increases in PAD-PCWP gradient in morbid obesity. We examined this possibility. Methods Included were 25 obese patients without daytime hypoxia undergoing bariatric surgery under general anesthesia. PvR was calculated as the difference between mean PA output pressure and PCWP divided by cardiac index. Mean PA output pressure was computed from the harmonic form of the recorded PA pressure by applying an attenuating factor to its phasic components, for which Fourier analysis was used. Total pulmonary vascular resistance (TPVR) was calculated as the difference between mean PA pressure and PCWP divided by cardiac index. To avoid the effect of PA resistance on TPVR and PvR, the PvR/TPVR ratio was used. Results There was a good correlation between PvR/TPVR ratio and PAD-PCWP gradient (r2=0.785, P<0.0001). When patients were divided into two groups based on PAD-PCWP gradient, the PvR/TPVR ratio was 0.67+/-0.06 (mean+/-SD) in the group with a PAD-PCWP gradient of at least 6 mmHg and 0.48+/-0.05 in the other group (P<0.0001). Conclusions A strong correlation between PvR/TPVR ratio and PAD-PCWP gradient suggests that the increased PvR contributes to increased PAD-PCWP gradient in obese patients without daytime hypoxia.

RADIATIVE LEPTONIC DECAY OF Ds AND D MESONS

The radiative leptonic decays Ds → μνμγ and D → μνμγ are investigated at the tree level within the relativistic independent quark model based on the confining potential in the scalar–vector harmonic form. The branching ratios for these decays in the vanishing lepton mass limit are obtained as ℬr(Ds → μνμγ) = 4.94×10-4 and ℬr(D → μνμγ) = 3.34×10-5, which includes the contributions of the internal bremsstrahlung and structure dependent diagrams at the level of the quark constituents. Finally, the photon energy spectra as predicted here is found to be symmetric about the peak value of the photon energy at [Formula: see text].

Some Algebraic and Geometric Structures on Poisson Manifolds

In this paper we study some algebraic properties of harmonic forms on Poisson manifolds. It is well known that in the classical case (on Riemannian manifolds) the product of harmonic forms is not harmonic. Here we describe the algebraic and analytical mechanisms explaining this fact. We also obtain a condition under which the product of de Rham cohomology classes, which includes harmonic representatives, can be represented by a harmonic form.

VANISHING THEOREM OF THE HODGE–KODAIRA OPERATOR FOR DIFFERENTIAL FORMS ON A CONVEX DOMAIN OF THE WIENER SPACE

We discuss the vanishing theorem on a convex domain of the Wiener space. We show that there is no harmonic form satisfying the absolute boundary condition. Our method relies on an expression of the bilinear form associated with the Hodge–Kodaira operator.

Harmony + Improvisation = Jazz

In the past, jazz musicians such as Miles Davis have had negative experiences of ‘straight’ academies and conservatories, and these institutions have been negative towards jazz. This may represent a conflict between creativity and recreativity. But as a teacher of jazz at the Guildhall School of Music the author is finding that this conflict disappears when students from jazz and classical backgrounds learn to improvise by the same approach. This approach works upwards from the harmonic basis of jazz, which in fact is the same as that of classical music. As, at the outset, both jazz and classical students often seem to lack a precise concept of underlying harmonic form, the author concludes that more needs to be done with harmony at an earlier stage in music education, and that jazz may be the best context in which for this to happen.

The Poincaré Dual of a Geodesic Algebraic Curve in a Quotient of the 2-Ball

We shall consider an irreducible, non-singular, totally geodesic holomorphic curve N in a compact quotient M = Γ\D of the unit ball D = {(z, w):|z|2 + |w|2 < 1} in C2 with the Kahler structure provided by the Bergman metric. The main result of this paper is an explicit construction of the harmonic form of type (1,1) which is dual to N. Our construction is as follows. Let p:D → Γ\D be the universal covering map. Choose a component D1 in the inverse image of N under p. The choice of D1 corresponds to choosing an embedding of the fundamental group of N into Γ. We denote the image by Γ1. Let π : D → D1 be the fiber bundle obtained by exponentiating the normal bundle of D1 in D. Let μ be the volume form of D1.