Classical Casimir free energy for two Drude spheres of arbitrary radii: A plane-wave approach

We derive an exact analytic expression for the high-temperature limit of the Casimir interaction between two Drude spheres of arbitrary radii. Specifically, we determine the Casimir free energy by using the scattering approach in the plane-wave basis. Within a round-trip expansion, we are led to consider the combinatorics of certain partitions of the round trips. The relation between the Casimir free energy and the capacitance matrix of two spheres is discussed. Previously known results for the special cases of a sphere-plane geometry as well as two spheres of equal radii are recovered. An asymptotic expansion for small distances between the two spheres is determined and analytical expressions for the coefficients are given.

Expected Maximum Block Size in Critical Random Graphs

Let G(n,M) be a uniform random graph with n vertices and M edges. Let ${\wp_{n,m}}$ be the maximum block size of G(n,M), that is, the maximum size of its maximal 2-connected induced subgraphs. We determine the expectation of ${\wp_{n,m}}$ near the critical point M = n/2. When n − 2M ≫ n2/3, we find a constant c1 such that $$c_1 = \lim_{n \rightarrow \infty} \left({1 - \frac{2M}{n}} \right) \,\E({\wp_{n,m}}).$$ Inside the window of transition of G(n,M) with M = (n/2)(1 + λn−1/3), where λ is any real number, we find an exact analytic expression for $$c_2(\lambda) = \lim_{n \rightarrow \infty} \frac{\E{\left({\wp_{n,{{(n/2)}({1+\lambda n^{-1/3}})}}}\right)}}{n^{1/3}}.$$ This study relies on the symbolic method and analytic tools from generating function theory, which enable us to describe the evolution of $n^{-1/3}\,\E{\left({\wp_{n,{{(n/2)}({1+\lambda n^{-1/3}})}}}\right)}$ as a function of λ.

The Stochastic Resonance Behaviors of a Generalized Harmonic Oscillator Subject to Multiplicative and Periodically Modulated Noises

The stochastic resonance (SR) characteristics of a generalized Langevin linear system driven by a multiplicative noise and a periodically modulated noise are studied (the two noises are correlated). In this paper, we consider a generalized Langevin equation (GLE) driven by an internal noise with long-memory and long-range dependence, such as fractional Gaussian noise (fGn) and Mittag-Leffler noise (M-Ln). Such a model is appropriate to characterize the chemical and biological solutions as well as to some nanotechnological devices. An exact analytic expression of the output amplitude is obtained. Based on it, some characteristic features of stochastic resonance phenomenon are revealed. On the other hand, by the use of the exact expression, we obtain the phase diagram for the resonant behaviors of the output amplitude versus noise intensity under different values of system parameters. These useful results presented in this paper can give the theoretical basis for practical use and control of the SR phenomenon of this mathematical model in future works.

Nonlinear Coupled Hydrostatics of Arctic Conical Platforms

The increased exploration of deeper Arctic waters motivates the designs of new floating structures to operate under harsh Arctic conditions. Based on several model tests and investigations, structures with conical sections at the waterline have been shown to be a good design for waters where drifting ice is present, because the approaching ice fails in bending, which induces smaller loads than a crushing failure of ice. However, in most Arctic waters ice features are only present during part of the year and a large portion of the operation time of these structures will be in open water. Therefore, the floating structures must perform well in both conditions.Conical sections at the waterline will induce nonlinear coupling in the hydrostatic restoring forces and moments. It is important to understand how this affects the behavior in both ice and open water conditions. In order to investigate the nonlinear coupled hydrostatic restoring forces, an exact analytic expression for the metacentric height of a regular cone is presented. This is further used to develop an exact analytic expression for the hydrostatic restoring forces and moments for any body whose waterline intersects the frustum of a cone. A platform of the shallow draft-type, the platform type for which exact hydrostatics is most important, is used as a basis for the discussion and the effect of the coupled nonlinear restoring forces is illustrated by comparison to a model test performed in both open water and ice conditions.

Eigenmodes of lined flow ducts with rigid splices

This paper presents an analytic expression for the acoustic eigenmodes of a cylindrical lined duct with rigid axially running splices in the presence of flow. The cylindrical duct is considered to be uniformly lined except for two symmetrically positioned axially running rigid liner splices. An exact analytic expression for the acoustic pressure eigenmodes is given in terms of an azimuthal Fourier sum, with the Fourier coefficients given by a recurrence relation. Since this expression is derived using a Green’s function method, the completeness of the expansion is guaranteed. A numerical procedure is described for solving this recurrence relation, which is found to converge exponentially with respect to number of Fourier terms used and is in practice quick to compute; this is then used to give several numerical examples for both uniform and sheared mean flow. An asymptotic expression is derived to directly calculate the pressure eigenmodes for thin splices. This asymptotic expression is shown to be quantitatively accurate for ducts with very thin splices of less than 1 % unlined area and qualitatively helpful for thicker splices of the order of 6 % unlined area. A thin splice is in some cases shown to increase the damping of certain acoustic modes. The influences of thin splices and thin boundary layers are compared and found to be of comparable magnitude for the parameters considered. Trapped modes at the splices are also identified and investigated.

A PRECISE DESCRIPTION OF THE p-ADIC VALUATION OF THE NUMBER OF ALTERNATING SIGN MATRICES

Following Sun and Moll ([4]), we study vp(T(N)), the p-adic valuation of the counting function of the alternating sign matrices. We find an exact analytic expression for it that exhibits the fluctuating behavior, by means of Fourier coefficients. The method is the Mellin–Perron technique, which is familiar in the analysis of the sum-of-digits function and related quantities.

Exact expression for the effective acoustics of patchy-saturated rocks

Seismic effects of a partially gas-saturated subsurface have been known for many years. For example, patches of nonuniform saturation occur at the gas-oil and gas-water contacts in hydrocarbon reservoirs. Open-pore boundary conditions are applied to the quasi-static Biot equations of poroelasticity to derive an exact analytic expression of the effective bulk modulus for partially saturated media with spherical gas patches larger than the typical pore size. The pore fluid and the rock properties can have different values in the central sphere and in the surrounding region. An analytic solution prevents loss of accuracy from ill-conditioned equations as encountered in the numerical solution for certain input. For a sandstone saturated with gas and water, we found that the P-wave velocity and attenuation in conventional models differ as much as 15% from the exact solution at seismic frequencies. This makes the use of present exact theory necessary to describe patchy saturation, although (more realistic) complex patch shapes and distributions were not considered. We found that, despite earlier corrections, the White conventional model does not yield the correct low-frequency asymptote for the attenuation.

Nonlinear Coupled Hydrostatics of Floating Conical Platforms

The increased exploration of deeper Arctic waters motivates the designs of new floating structures to operate under harsh Arctic conditions. Based on several model tests and investigations, structures with conical sections at the waterline have been shown to be a good design for waters where drifting ice is present, because the approaching ice fails in bending which induces smaller loads than crushing failure. However, in most Arctic waters ice features are only present parts of the year and a large portion of the operation time of these structures will be in open water. Therefore, the floating structures must perform well in both these conditions. Conical sections at the waterline will induce nonlinear coupling in the hydrostatic restoring forces and moments. It is important to understand how this affects the behaviour in both ice and open water conditions. In order to investigate the nonlinear coupled hydrostatic restoring forces, an exact analytic expression for the metacentric height of a regular cone is presented. This is further derived to a frustum cone that can be used to develop an exact analytic expression for the hydrostatic restoring forces and moments for any conical body. A platform of the shallow draught-type is used as a basis for the discussion and the effect of the coupled nonlinear restoring forces is illustrated by two-degrees-of-freedom (2DOF) pitch-heave time domain simulations.

ENERGY EXTRACTION FROM GRAVITATIONAL COLLAPSE TO STATIC BLACK HOLES

The mass-energy formula of black holes implies that up to 50% of the energy can be extracted from a static black hole. Such a result is reexamined using the recently established analytic formulas for the collapse of a shell and the expression for the irreducible mass of a static black hole. It is shown that the efficiency of energy extraction process during the formation of the black hole is linked in an essential way to the gravitational binding energy, the formation of the horizon and the reduction of the kinetic energy of implosion. Here a maximum efficiency of 50% in the extraction of the mass energy is shown to be generally attainable in the collapse of a spherically symmetric shell: surprisingly this result holds as well in the two limiting cases of the Schwarzschild and extreme Reissner–Nordström space–times. Moreover, the analytic expression recently found for the implosion of a spherical shell to an already formed black hole leads to a new exact analytic expression for the energy extraction which results in an efficiency strictly less than 100% for any physical implementable process. There appears to be no incompatibility between General Relativity and Thermodynamics at this classical level.

CHOPTUIK SCALING AND QUASINORMAL MODES IN THE ADS/CFT CORRESPONDENCE

The exact quasinormal frequencies for scalar perturbations of the three-dimensional BTZ black hole were presented1. The timescale for the decay of the scalar perturbation is given by the imaginary part of the quasinormal frequencies. Via the AdS/CFT correspondence, one then obtains a prediction of the timescale for return to equilibrium of the dual conformal field theory. By studying a two-particle collision process, an exact analytic expression for the Choptuik scaling parameter of the BTZ black hole was obtained2. A possible connection between the Choptuik scaling parameter and the imaginary part of the quasinormal frequencies was explored1.