On a general solution of Hill's equation

i. ON A GENERAL SOLUTION OF HILL'S EQUATION. BY E. LINDSAY INCE. Reprinted from the MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY, Vol. LXIV. No. 5. • ii. NOTES ON THE GENERAL SOLUTION OF HILL'S EQUATION. BY E. LINDSAY INCE. Reprinted from the MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY, Vol. LXXVI. No. 5. • iii. FURTHER NOTES ON THE GENERAL SOLUTION OF HILL'S EQUATION. BY E. LINDSAY INCE. Reprinted from the MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY, Vol. LXXVIII. No. 2. • iv. On the continued fractions associated with the hypergenerative equatio

On \mu-Compatible Metrics and Measurable Sensitivity

We introduce the notion of W-measurable sensitivity, which extends and strictly implies canonical measurable sensitivity, a measure- theoretic version of sensitive dependence on initial conditions. This notion also implies pairwise sensitivity with respect to a large class of metrics. We show that nonsingular ergodic and conservative dynamical systems on standard spaces must be either W-measurably sensitive, or isomorphic mod 0 to a minimal uniformly rigid isometry. In the finite measure-preserving case they are W-measurably sensitive or measurably isomorphic to an ergodic isometry on a compact metric space.Comment: Many improvements in exposition, a technical assumption removed, as suggested by the reviewe

Penrose Limit and String Theories on Various Brane Backgrounds

We investigate the Penrose limit of various brane solutions including Dp-branes, NS5-branes, fundamental strings, (p,q) fivebranes and (p,q) strings. We obtain special null geodesics with the fixed radial coordinate (critical radius), along which the Penrose limit gives string theories with constant mass. We also study string theories with time-dependent mass, which arise from the Penrose limit of the brane backgrounds. We examine equations of motion of the strings in the asymptotic flat region and around the critical radius. In particular, for (p,q) fivebranes, we find that the string equations of motion in the directions with the B field are explicitly solved by the spheroidal wave functions.Comment: 41 pages, Latex, minor correction

It\u27s a \u27Criming Shame\u27: Moving from Land Use Ethics to Criminalization of Behavior Leading to Permits and Other Zoning Related Acts

In the past, land use ethics inquiries predominately involved conflicts of interest or an official holding public office while engaging in a previously held business or law practice. Now, prosecutors are looking at the underlying criminality of the unethical acts carried out in the context of land use decisions. With a wide array of criminal statutes in the hands of federal prosecutors, almost all forms of unethical conduct could in some way also violate a federal criminal statute.Part II of this article reviews the federal statutes most often used by federal prosecutors and provides some examples of recent reported cases in which the underlying illegal or unethical conduct involved alleged criminal activity. Part III offers some examples of recent reported state court cases in which criminal acts involving land use permitting or decision-making were the underlying cause of the subsequent or reported court action. Part IV concludes with the caveat that municipal attorneys and public officials can no longer simply view ethical issues in land use as a local or state civil matter, and those who work in and advise those in the public sector should be mindful of the tools at the disposal of federal investigators and prosecutors

It\u27s a \u27Criming Shame\u27: Moving from Land Use Ethics to Criminalization of Behavior Leading to Permits and Other Zoning Related Acts

In the past, land use ethics inquiries predominately involved conflicts of interest or an official holding public office while engaging in a previously held business or law practice. Now, prosecutors are looking at the underlying criminality of the unethical acts carried out in the context of land use decisions. With a wide array of criminal statutes in the hands of federal prosecutors, almost all forms of unethical conduct could in some way also violate a federal criminal statute.Part II of this article reviews the federal statutes most often used by federal prosecutors and provides some examples of recent reported cases in which the underlying illegal or unethical conduct involved alleged criminal activity. Part III offers some examples of recent reported state court cases in which criminal acts involving land use permitting or decision-making were the underlying cause of the subsequent or reported court action. Part IV concludes with the caveat that municipal attorneys and public officials can no longer simply view ethical issues in land use as a local or state civil matter, and those who work in and advise those in the public sector should be mindful of the tools at the disposal of federal investigators and prosecutors

Helical Magnetorotational Instability in Magnetized Taylor-Couette Flow

Hollerbach and Rudiger have reported a new type of magnetorotational instability (MRI) in magnetized Taylor-Couette flow in the presence of combined axial and azimuthal magnetic fields. The salient advantage of this "helical'' MRI (HMRI) is that marginal instability occurs at arbitrarily low magnetic Reynolds and Lundquist numbers, suggesting that HMRI might be easier to realize than standard MRI (axial field only). We confirm their results, calculate HMRI growth rates, and show that in the resistive limit, HMRI is a weakly destabilized inertial oscillation propagating in a unique direction along the axis. But we report other features of HMRI that make it less attractive for experiments and for resistive astrophysical disks. Growth rates are small and require large axial currents. More fundamentally, instability of highly resistive flow is peculiar to infinitely long or periodic cylinders: finite cylinders with insulating endcaps are shown to be stable in this limit. Also, keplerian rotation profiles are stable in the resistive limit regardless of axial boundary conditions. Nevertheless, the addition of toroidal field lowers thresholds for instability even in finite cylinders.Comment: 16 pages, 2 figures, 1 table, submitted to PR

Nonlocal symmetries of Riccati and Abel chains and their similarity reductions

We study nonlocal symmetries and their similarity reductions of Riccati and Abel chains. Our results show that all the equations in Riccati chain share the same form of nonlocal symmetry. The similarity reduced $N^{th}$ order ordinary differential equation (ODE), $N=2, 3,4,...$, in this chain yields $(N-1)^{th}$ order ODE in the same chain. All the equations in the Abel chain also share the same form of nonlocal symmetry (which is different from the one that exist in Riccati chain) but the similarity reduced $N^{th}$ order ODE, $N=2, 3,4,$, in the Abel chain always ends at the $(N-1)^{th}$ order ODE in the Riccati chain. We describe the method of finding general solution of all the equations that appear in these chains from the nonlocal symmetry.Comment: Accepted for publication in J. Math. Phy

Extension of Nikiforov-Uvarov Method for the Solution of Heun Equation

We report an alternative method to solve second order differential equations which have at most four singular points. This method is developed by changing the degrees of the polynomials in the basic equation of Nikiforov-Uvarov (NU) method. This is called extended NU method for this paper. The eigenvalue solutions of Heun equation and confluent Heun equation are obtained via extended NU method. Some quantum mechanical problems such as Coulomb problem on a 3-sphere, two Coulombically repelling electrons on a sphere and hyperbolic double-well potential are investigated by this method

Periodic solutions of a many-rotator problem in the plane. II. Analysis of various motions

Various solutions are displayed and analyzed (both analytically and numerically) of arecently-introduced many-body problem in the plane which includes both integrable and nonintegrable cases (depending on the values of the coupling constants); in particular the origin of certain periodic behaviors is explained. The light thereby shone on the connection among \textit{integrability} and \textit{analyticity} in (complex) time, as well as on the emergence of a \textit{chaotic} behavior (in the guise of a sensitive dependance on the initial data) not associated with any local exponential divergence of trajectories in phase space, might illuminate interesting phenomena of more general validity than for the particular model considered herein.Comment: Published by JNMP at http://www.sm.luth.se/math/JNMP