Some climatological characteristics of the changing pattern of temperature in Bangladesh – A diagnostic approach

Based on climatological data of maximum and minimum temperatures of seventeen stations for a period of 60 years (1949 - 2008), obtained from Bangladesh Meteorological Department (BMD), the changing pattern of temperature in Bangladesh has been investigated. Some characteristics of annual variation and spatial distribution of mean monthly minimum, mean monthly maximum and mean monthly temperature, annual variation and spatial distribution of mean monthly amplitude of temperature have been explained. The characteristics of changing pattern of temperature such as standard deviation, coefficient of variation, ratio between mean monthly maximum and mean monthly minimum temperatures have been studied. The run of the temperature in different stations for the years 1949-2008, the periodicity of mean annual temperature and mean decade deviations have been discussed. The integral curve of mean annual temperature for Dhaka has been prepared and explained. Regression analysis for mean monthly temperature of January, April, July and October for the stations Sylhet, Chittagong, Khulna and Rangpur with Dhaka have been analyzed. Some linear correlation equations have been deduced. In the investigation, attempts (as far as possible) have been made to explain the synoptic cause of the changing pattern of temperature in Bangladesh.

Explicit methods for integrating stiff Cauchy problems

An explicit method for solving stiff Cauchy problems is proposed. The method relies on explicit schemes and a step size selection algorithm based on the curvature of an integral curve. Closed-form formulas are derived for finding the curvature. For Runge-Kutta schemes with up to four stages, the corresponding sets of coefficients are given. The method is validated on a test problem with a given exact solution. It is shown that the method is as accurate and robust as implicit methods, but is substantially superior to them in efficiency. A numerical example involving chemical kinetics computations with 9 components and 50 reactions is given.

Vector Fields

The velocity of a fluid at each point of space-time is a vector field (or flow). It is best to think of it in terms of the effect of fluid flow on some scalar field. A vector field is thus a first order partial differential operator, called the material derivative in fluid mechanics. The path of a speck of dust carried along (advected) by the fluid is the integral curve of the velocity field. Even simple vector fields can have quite complicated integral curves: a manifestation of chaos. Of special interest are incompressible (with zero divergence) and irrotational (with zero curl) flows. A fixed point of a vector field is a point at which it vanishes. The derivative of a vector field at a fixed point is a matrix (the Jacobi matrix) whose spectrum is independent of the choice of coordinates.

Motion of an integral curve of a Hamiltonian dynamical system and the evolution equations in 3D

We show that all of the nonstretching curve motions specified in the Frenet–Serret frame in the literature can be described by the time evolution of an integral curve of a Hamiltonian dynamical system such that the underlying curve is a geodesic curve on a leaf of the foliation determined by the Poisson structure in three dimensions. As an illustrative example, we show that the focusing version of the nonlinear Schrödinger equation and the complex modified Korteweg–de Vries (mKdV) equation are obtained in this way.

An application of N-Bishop frame to spherical images for direction curves

In this paper, we first introduce [Formula: see text]-Bishop frame for a normal direction curve which is defined as an integral curve of the principal normal of a curve. We express this new frame and its properties. Afterwards, we obtain new spherical images by translating [Formula: see text]-Bishop frame vectors to the center of unit sphere [Formula: see text] in [Formula: see text]. Then, these new spherical images are called [Formula: see text]-Bishop spherical images. Additionally, we compute the Frénet–Serret equations of these new spherical images. Moreover, we show that integral curves of [Formula: see text]-Bishop spherical images of slant helices are also slant helices. Finally, we present some illustrated examples.

On stabilizers of algebraic function fields of one variable

Let $\tilde K$ be a fixed algebraic closure of an infinite field K. We consider an absolutely integral curve Γ in $\mathbb{P}_{K}^{n}$ with n ≥ 2. The curve $\it\Gamma_{\tilde{K}}$ should have only finitely many inflection points, finitely many double tangents, and there exists no point in $\mathbb{P}_{\tilde{K}}^{n}$ through which infinitely many tangents to $\it\Gamma_{\tilde{K}}$ go. In addition there exists a prime number q such that $\it\Gamma_{\tilde{K}}$ has a cusp of multiplicity q and the multiplicities of all other points of $\it\Gamma_{\tilde{K}}$ are at most q. Under these assumptions, we construct a non-empty Zariski-open subset O of $\mathbb{P}_{\tilde{K}}^{n}$ such that if n ≥ 3, the projection from each point o ∈ O(K) birationally maps Γ onto an absolutely integral curve Γ′ in $\mathbb{P}_{K}^{n-1}$ with the same properties as Γ (keeping q unchanged). If n = 2, then the projection from each o ∈ O(K) maps Γ onto $\mathbb{P}_{K}^{1}$ and leads to a stabilizing element t of the function field F of Γ over K. The latter means that F/K(t) is a finite separable extension whose Galois closure ${\hat F}$ is regular over K.

Projective curves of degree=codimension+2 II

Let [Formula: see text] be a nondegenerate projective integral curve of degree [Formula: see text] which is not linearly normal. In this paper, we continues the study begun in [E. Park, Projective curves of degree=codimension+2, Math. Z. 256 (2007) 685–697] for the minimal free resolution of [Formula: see text]. It is well-known that [Formula: see text] is an isomorphic projection of a rational normal curve [Formula: see text] from a point [Formula: see text]. Our main result is about how the graded Betti numbers of [Formula: see text] are determined by the rank of [Formula: see text] with respect to [Formula: see text], which is a measure of the relative location of [Formula: see text] from [Formula: see text].