A nodal domain theorem for integrable billiards in two dimensions

Eigenfunctions of integrable planar billiards are studied - in particular, the number of nodal domains, nu of the eigenfunctions with Dirichlet boundary conditions are considered. The billiards for which the time-independent Schrodinger equation (Helmholtz equation) is separable admit trivial expressions for the number of domains. Here, we discover that for all separable and nonseparable integrable billiards, nu satisfies certain difference equations. This has been possible because the eigenfunctions can be classified in families labelled by the same value of m mod kn, given a particular k, for a set of quantum numbers, m, n. Further, we observe that the patterns in a family are similar and the algebraic representation of the geometrical nodal patterns is found. Instances of this representation are explained in detail to understand the beauty of the patterns. This paper therefore presents a mathematical connection between integrable systems and difference equations. (C) 2014 Elsevier Inc. All rights reserved.

A gap theorem for Ricci-flat 4-manifolds

Let (M, g) be a compact Ricci-fiat 4-manifold. For p is an element of M let K-max(P) (respectively K-min(p)) denote the maximum (respectively the minimum) of sectional curvatures at p. We prove that if K-max(p) <= -cK(min)(P) for all p is an element of M, for some constant c with 0 <= c < 2+root 6/4 then (M, g) is fiat. We prove a similar result for compact Ricci-flat Kahler surfaces. Let (M, g) be such a surface and for p is an element of M let H-max(p) (respectively H-min(P)) denote the maximum (respectively the minimum) of holomorphic sectional curvatures at p. If H-max(P) <= -cH(min)(P) for all p is an element of M, for some constant c with 0 <= c < 1+root 3/2, then (M, g) is flat. (C) 2015 Elsevier B.V. All rights reserved.

FAST AND ACCURATE BILATERAL FILTERING USING GAUSS-POLYNOMIAL DECOMPOSITION

The bilateral filter is a versatile non-linear filter that has found diverse applications in image processing, computer vision, computer graphics, and computational photography. A common form of the filter is the Gaussian bilateral filter in which both the spatial and range kernels are Gaussian. A direct implementation of this filter requires O(sigma(2)) operations per pixel, where sigma is the standard deviation of the spatial Gaussian. In this paper, we propose an accurate approximation algorithm that can cut down the computational complexity to O(1) per pixel for any arbitrary sigma (constant-time implementation). This is based on the observation that the range kernel operates via the translations of a fixed Gaussian over the range space, and that these translated Gaussians can be accurately approximated using the so-called Gauss-polynomials. The overall algorithm emerging from this approximation involves a series of spatial Gaussian filtering, which can be efficiently implemented (in parallel) using separability and recursion. We present some preliminary results to demonstrate that the proposed algorithm compares favorably with some of the existing fast algorithms in terms of speed and accuracy.

A role for nocturnal serum testosterone surge in regulating spermatogenesis in the adult non-human primate

Adult male bonnet monkeys exhibit nychthemeral rhythms in testosterone (T) secretion but the precise role of this heightened level of T secretion in regulating spermatogenesis is not known. Intranasal administration of microdoses (500 mu g or 250 mu g/day) of Norethisterone (IN-NET) to adult monkeys (n = 6) at 1600 h each day selectively and completely suppressed the nocturnal surge levels of serum T. Concomitant with this was a significant reduction (P<0.01) in serum LH but not FSH levels. DNA flow cytometric analysis of testicular biopsy tissue showed by week 10 of IN-NET treatment an arrest in meiotic transformation of primary spermatocytes (4C) to round/elongate (1C/HC) spermatids and by week 20 there was a complete absence of 4C, 1C and HC cells (with a relative accumulation in 2C cells). The accumulated meiotic (4C) cells at week 10 showed an increase (>80%, P<0.01) in coefficient of variation and a decrease in intensity of DNA-bound ethidium bromide fluorescence, parameters characteristic of degenerating 'apoptotic' subpopulation of germ cells. While two monkeys exhibited acute oligozoospermia 4 became azoospermic by 20 weeks of IN-NET treatment. A complete, qualitative reversal in the regressive changes in spermatogenesis and near-normal sperm output were apparent at the end of a 20-week recovery phase. These data demonstrate that prolonged, selective suppression of nocturnal surge levels of serum T secretion exerts a primary effect on meiosis in spermatogenesis leading to oligo/azoospermic status in adult bonnet monkeys.

Assessment of luteal rescue and desensitization of macaque corpus luteum brought about by human chorionic gonadotrophin and deglycosylated human chorionic gonadotrophin treatment

The objective of the current study was to investigate the mechanism by which the corpus luteum (CL) of the monkey undergoes desensitization to luteinizing hormone following exposure to increasing concentration of human chorionic gonadotrophin (hCG) as it occurs in pregnancy. Female bonnet monkeys were injected (im) increasing doses of hCG or dghCG beginning from day 6 or 12 of the luteal phase for either 10 or 4 or 2 days. The day of oestrogen surge was considered as day '0' of luteal phase. Luteal cells obtained from CL of these animals were incubated with hCG (2 and 200 pg/ml) or dbcAMP (2.5, 25 and 100 mu M) for 3 h at 37 degrees C and progesterone secreted was estimated. Corpora lutea of normal cycling monkeys on day 10/16/22 of the luteal phase were used as controls, In addition the in vivo response to CG and deglycosylated hCG (dghCG) was assessed by determining serum steroid profiles following their administration. hCG (from 15-90 IU) but not dghCG (15-90 IU) treatment in vivo significantly (P < 0.05) elevated serum progesterone and oestradiol levels. Serum progesterone, however, could not be maintained at a elevated level by continuous treatment with hCG (from day 6-15), the progesterone level declining beyond day 13 of luteal phase. Administering low doses of hCG (15-90 IU/day) from day 6-9 or high doses (600 IU/day) on days 8 and 9 of the luteal phase resulted in significant increase (about 10-fold over corresponding control P < 0.005) in the ability of luteal cells to synthesize progesterone (incubated controls) in vitro. The luteal cells of the treated animals responded to dbcAMP (P < 0.05) but not to hCG added in vitro, The in vitro response of luteal cells to added hCG was inhibited by 0, 50 and 100% if the animals were injected with low (15-90 IU) or medium (100 IU) between day 6-9 of luteal phase and high (600 IU on day 8 and 9 of luteal phase) doses of dghCG respectively; such treatment had no effect on responsivity of the cells to dbcAMP, The luteal cell responsiveness to dbcAMP in vitro was also blocked if hCG was administered for 10 days beginning day 6 of the luteal phase. Though short term hCG treatment during late luteal phase (from days 12-15) had no effect on luteal function, 10 day treatment beginning day 12 of luteal phase resulted in regain of in vitro responsiveness to both hCG (P < 0.05) and dbcAMP (P < 0.05) suggesting that luteal rescue can occur even at this late stage. In conclusion, desensitization of the CL to hCG appears to be governed by the dose/period for which it is exposed to hCG/dghCG. That desensitization is due to receptor occupancy is brought out by the fact that (i) this can be achieved by giving a larger dose of hCG over a 2 day period instead of a lower dose of the hormone for a longer (4 to 10 days) period and (ii) the effect can largely be reproduced by using dghCG instead of hCG to block the receptor sites. It appears that to achieve desensitization to dbcAMP also it is necessary to expose the luteal cell to relatively high dose of hCG for more than 4 days.

Implicit and multigrid procedures for steady-state computations with upwind algorithms

A computational study for the convergence acceleration of Euler and Navier-Stokes computations with upwind schemes has been conducted in a unified framework. It involves the flux-vector splitting algorithms due to Steger-Warming and Van Leer, the flux-difference splitting algorithms due to Roe and Osher and the hybrid algorithms, AUSM (Advection Upstream Splitting Method) and HUS (Hybrid Upwind Splitting). Implicit time integration with line Gauss-Seidel relaxation and multigrid are among the procedures which have been systematically investigated on an individual as well as cumulative basis. The upwind schemes have been tested in various implicit-explicit operator combinations such that the optimal among them can be determined based on extensive computations for two-dimensional flows in subsonic, transonic, supersonic and hypersonic flow regimes. In this study, the performance of these implicit time-integration procedures has been systematically compared with those corresponding to a multigrid accelerated explicit Runge-Kutta method. It has been demonstrated that a multigrid method employed in conjunction with an implicit time-integration scheme yields distinctly superior convergence as compared to those associated with either of the acceleration procedures provided that effective smoothers, which have been identified in this investigation, are prescribed in the implicit operator.

Bounds on isoperimetric values of trees

Let G = (V, E) be a finite, simple and undirected graph. For S subset of V, let delta(S, G) = {(u, v) is an element of E : u is an element of S and v is an element of V - S} be the edge boundary of S. Given an integer i, 1 <= i <= vertical bar V vertical bar, let the edge isoperimetric value of G at i be defined as b(e)(i, G) = min(S subset of V:vertical bar S vertical bar=i)vertical bar delta(S, G)vertical bar. The edge isoperimetric peak of G is defined as b(e)(G) = max(1 <= j <=vertical bar V vertical bar)b(e)(j, G). Let b(v)(G) denote the vertex isoperimetric peak defined in a corresponding way. The problem of determining a lower bound for the vertex isoperimetric peak in complete t-ary trees was recently considered in [Y. Otachi, K. Yamazaki, A lower bound for the vertex boundary-width of complete k-ary trees, Discrete Mathematics, in press (doi: 10.1016/j.disc.2007.05.014)]. In this paper we provide bounds which improve those in the above cited paper. Our results can be generalized to arbitrary (rooted) trees. The depth d of a tree is the number of nodes on the longest path starting from the root and ending at a leaf. In this paper we show that for a complete binary tree of depth d (denoted as T-d(2)), c(1)d <= b(e) (T-d(2)) <= d and c(2)d <= b(v)(T-d(2)) <= d where c(1), c(2) are constants. For a complete t-ary tree of depth d (denoted as T-d(t)) and d >= c log t where c is a constant, we show that c(1)root td <= b(e)(T-d(t)) <= td and c(2)d/root t <= b(v) (T-d(t)) <= d where c(1), c(2) are constants. At the heart of our proof we have the following theorem which works for an arbitrary rooted tree and not just for a complete t-ary tree. Let T = (V, E, r) be a finite, connected and rooted tree - the root being the vertex r. Define a weight function w : V -> N where the weight w(u) of a vertex u is the number of its successors (including itself) and let the weight index eta(T) be defined as the number of distinct weights in the tree, i.e eta(T) vertical bar{w(u) : u is an element of V}vertical bar. For a positive integer k, let l(k) = vertical bar{i is an element of N : 1 <= i <= vertical bar V vertical bar, b(e)(i, G) <= k}vertical bar. We show that l(k) <= 2(2 eta+k k)

Separability in relativistic Hamiltonian particle dynamics

The problem of separability in recent models of classical relativistic interacting particles is examined. This physical requirement is shown to be more subtle than naive separability of all the constraints defining the system: it is adequate to be able to canonically transform the time-fixing constraints from an unseparated to a separated form when clusters emerge. Viewing separability in this way, and within a specific framework, we are led to a new no-interaction theorem which states the incompatibility of nontrivial interaction with relativistic invariance, separability, and invariant world lines for more than two particles.

Do H-functions always increase during Violent Relaxation

Recent work on the violent relaxation of collisionless stellar systems has been based on the notion of a wide class of entropy functions. A theorem concerning entropy increase has been proved. We draw attention to some underlying assumptions that have been ignored in the applications of this theorem to stellar dynamical problems. Once these are taken into account, the use of this theorem is at best heuristic. We present a simple counter-example.

Classical Langevin dynamics of a charged particle moving on a sphere and diamagnetism: A surprise

It is generally known that the orbital diamagnetism of a classical system of charged particles in thermal equilibrium is identically zero —the Bohr-van Leeuwen theorem. Physically, this null result derives from the exact cancellation of the orbital diamagnetic moment associated with the complete cyclotron orbits of the charged particles by the paramagnetic moment subtended by the incomplete orbits skipping the boundary in the opposite sense. Motivated by this crucial but subtle role of the boundary, we have simulated here the case of a finite but unbounded system, namely that of a charged particle moving on the surface of a sphere in the presence of an externally applied uniform magnetic field. Following a real space-time approach based on the classical Langevin equation, we have computed the orbital magnetic moment that now indeed turns out to be non-zero and has the diamagnetic sign. To the best of our knowledge, this is the first report of the possibility of finite classical diamagnetism in principle, and it is due to the avoided cancellation.

Asymptotic behaviour of point processes with Markov-dependent intervals

This splitting techniques for MARKOV chains developed by NUMMELIN (1978a) and ATHREYA and NEY (1978b) are used to derive an imbedded renewal process in WOLD's point process with MARKOV-correlated intervals. This leads to a simple proof of renewal theorems for such processes. In particular, a key renewal theorem is proved, from which analogues to both BLACKWELL's and BREIMAN's forms of the renewal theorem can be deduced.

Forms of relativistic dynamics with World Line Condition and separability

he Dirac generator formalism for relativistic Hamiltonian dynamics is reviewed along with its extension to constraint formalism. In these theories evolution is with respect to a dynamically defined parameter, and thus time evolution involves an eleventh generator. These formulations evade the No-Interaction Theorem. But the incorporation of separability reopens the question, and together with the World Line Condition leads to a second no-interaction theorem for systems of three or more particles. Proofs are omitted, but the results of recent research in this area is highlighted.

On the invariants of coloured Petri Nets

In this paper, we develop a theorem that enables computation of the place invariants of the union of a finite collection of coloured Petri Nets when the individual nets satisfy certain conditions and their invariants are known. We consider the illustrative examples of the Readers-Writers problem, a resource sharing system, and a network of databases and show how this theorem is a valuable tool in the analysis of concurrent systems.

Spontaneous symmetry breaking in twisted noncommutative quantum theories

We analyze aspects of symmetry breaking for Moyal spacetimes within a quantization scheme which preserves the twisted Poincare´ symmetry. Towards this purpose, we develop the Lehmann-Symanzik- Zimmermann (LSZ) approach for Moyal spacetimes. The latter gives a formula for scattering amplitudes on these spacetimes which can be obtained from the corresponding ones on the commutative spacetime. This formula applies in the presence of spontaneous breakdown of symmetries as well. We also derive Goldstone’s theorem on Moyal spacetime. The formalism developed here can be directly applied to the twisted standard model.

A Novel Operator-Splitting Technique for One-Dimensional Laminar Premixed Flames

This paper is concerned the calculation of flame structure of one-dimensional laminar premixed flames using the technique of operator-splitting. The technique utilizes an explicit method of solution with one step Euler for chemistry and a novel probabilistic scheme for diffusion. The relationship between diffusion phenomenon and Gauss-Markoff process is exploited to obtain an unconditionally stable explicit difference scheme for diffusion. The method has been applied to (a) a model problem, (b) hydrazine decomposition, (c) a hydrogen-oxygen system with 28 reactions with constant Dρ 2 approximation, and (d) a hydrogen-oxygen system (28 reactions) with trace diffusion approximation. Certain interesting aspects of behaviour of the solution with non-unity Lewis number are brought out in the case of hydrazine flame. The results of computation in the most complex case are shown to compare very favourably with those of Warnatz, both in terms of accuracy of results as well as computational time, thus showing that explicit methods can be effective in flame computations. Also computations using the Gear-Hindmarsh for chemistry and the present approach for diffusion have been carried out and comparison of the two methods is presented.