Singular Discs Research Articles

A new algebraic solution for transforming Cartesian to geodetic coordinates

An exact and stable algebraic solution based on solving a quartic equation with respect to the cosine function of the reduced latitude is proposed to transform Cartesian into geodetic coordinates. The unique proper root of the equation appropriate to the transformation is chosen from all possible roots by rigorous analyses and the singular region of the transformation that in which there at least one component of the geodetic coordinates is indeterminate or non-single-valued characteristics are determined strictly. The new algorithm does not need any approximation and the instability problems incurred in other algebraic solutions are overcome. For practical applications, the algorithm performs comparably to that of [Vermeille, H., 2011. An analytical method to transform geocentric into geodetic coordinates. Journal of geodesy, 85 (2), 105–117.] and shows a certain superiority in the singular disc over Vermeille's algorithm.

Analytical and Semi-Analytical Formulas for the Self and Mutual Inductances of Concentric Coplanar Ordinary and Bitter Disk Coils

In this paper, analytical and semi-analytical formulas are presented for the self- and mutual inductance of thin ordinary disk coils and thin Bitter disk coils. The coils lie concentrically in a plane. The ordinary coils are coils with constant current density. The current density of a current carrying Bitter disc is not uniform across its cross-sectional area, but it is a function of the ratio of the inner diameter of the disk to an arbitrary radius within the disk. In this paper, we show the possibility to calculate the mutual and self-inductance of thin disk coils from the real coils of the cross-sections using some valuable conditions. The formulas for the mutual inductance and the self-inductance were obtained in the semi-analytic form as the combination of the elliptic integral of the second kind and a simple integral for the ordinary disk coils. The mutual inductance and self-inductance were obtained in the analytical form as the elliptic integral of the second kind for the Bitter disk coils. The formula for the self-inductance of the ordinary full disk was obtained in the close form. All formulas are given in remarkably simple form and give perfectly accurate results with a significantly small computational time. All cases of either regular or singular (disks in contact or overlapping) are covered. Many presented examples show the excellent numerical agreement with previously published methods.

Weighted surface algebras: General version

We introduce the general weighted surface algebras of triangulated surfaces with arbitrarily oriented triangles and describe their basic properties. In particular, we prove that all these algebras, except the singular disc, triangle, tetrahedral and spherical algebras, are symmetric tame periodic algebras of period 4.

Silver nano fabrication using leaf disc of Passiflora foetida Linn

The main purpose of the experiment is to develop a greener low cost SNP fabrication steps using factories of secondary metabolites from Passiflora leaf extract. Here, the leaf extraction process is omitted, and instead a leaf disc was used for stable SNP fabricated by optimizing parameters such as a circular leaf disc of 2 cm (1, 2, 3, 4, 5) instead of leaf extract and grade of pH (7, 8, 9, 11). The SNP synthesis reaction is tried under room temperature, sun, UV and dark condition. The leaf disc preparation steps are also discussed in details. The SNP obtained using (1 mM: 100 ml AgNO3+ singular leaf disc: pH 9, 11) is applied against featured room temperature and sun condition. The UV spectroscopic analysis confirms that sun rays synthesized SNP yields stable nano particles. The FTIR analysis confirms a large number of functional groups such as alkanes, alkyne, amines, aliphatic amine, carboxylic acid; nitro-compound, alcohol, saturated aldehyde and phenols involved in reduction of silver salt to zero valent ions. The leaf disc mediated synthesis of silver nanoparticles, minimizes leaf extract preparation step and eligible for stable SNP synthesis. The methods sun and room temperature based nano particles synthesized within 10 min would be use certainly for antimicrobial activity.

A characterization of the Γ-polynomials of knots with clasp number at most two

It is known that every knot bounds a singular disk with only clasp singularities, which is called a clasp disk. The clasp number of a knot is the minimum number of clasp singularities among all clasp disks of the knot. It is known that the Conway polynomials of knots with clasp number at most two are characterized. In this paper, we focus on the common zeroth coefficient polynomial of both the HOMFLYPT and Kauffman polynomials, which is called the [Formula: see text]-polynomial. As a result, we characterize the [Formula: see text]-polynomials of knots with clasp number at most two. Moreover, it is known that there exist two homeomorphic classes of clasp disks with two clasp singularities, which are called types [Formula: see text] and [Formula: see text]. We show that there exist infinitely many knots which bound clasp disks of not doubled knots but both types [Formula: see text] and [Formula: see text], not type [Formula: see text] but type [Formula: see text], not type [Formula: see text] but type [Formula: see text], respectively.

Approximation of Singular Discs for CR Extension

We introduce a class of singular discs, modelled on the one introduced by Zaitsev, Zampieri and the author. We show that they gain regularity from reparametrization and from taking powers. We set up a Bishop equation to attach these discs to a CR manifold with smooth dependence on parameters. By these tools we get extension of CR functions from wedges endowed with the sector property.

Structures in a class of magnetized scale-free discs

We construct analytically stationary global configurations for both aligned and logarithmic spiral coplanar magnetohydrodynamics (MHD) perturbations in an axisymmetric background MHD disc with a power-law surface mass density Σ 0 r -α , a coplanar azimuthal magnetic field B 0 oc r -γ , a consistent self-gravity and a power-law rotation curve v 0 oc r -β , where v 0 is the linear azimuthal gas rotation speed. The barotropic equation of state n Σ n is adopted for both MHD background equilibrium and coplanar MHD perturbations where n is the vertically integrated pressure and n is the barotropic index. For a scale-free background MHD equilibrium, a relation exists among a, β, y and n such that only one parameter (e.g. β) is independent. For a linear axisymmetric stability analysis, we provide global criteria in various parameter regimes. For non-axisymmetric aligned and logarithmic spiral cases, two branches of perturbation modes (i.e. fast and slow MHD density waves) can be derived once β is specified. To complement the magnetized singular isothermal disc analysis of Lou, we extend the analysis to a wider range of -1/4 < β < 1/2. As an illustrative example, we discuss specifically the β = 1/4 case when the background magnetic field is force-free. Angular momentum conservation for coplanar MHD perturbations and other relevant aspects of our approach are discussed.

Non-Axisymmetric Modes in Relativistic Singular Isothermal Disks

The relativistic singular isothermal disk equilibrium solutions are self-similar and form a two-parameter family described by the sound speed and the linear rotational velocity. Using the equilibrium model as the base state, we study its stability against self-similar but non-axisymmetric perturbations. We discovered that instability can be manifested through radiation driven neutral modes and bifurcation points to non-axisymmetric equilibria.

A natural framing of knots

Given a knot K in the 3-sphere, consider a singular disk bounded by K and the intersections of K with the interior of the disk. The absolute number of intersections, minimised over all choices of singular disk with a given algebraic number of intersections, defines the framing function of the knot. We show that the framing function is symmetric except at a finite number of points. The symmetry axis is a new knot invariant, called the natural framing of the knot. We calculate the natural framing of torus knots and some other knots, and discuss some of its properties and its relations to the signature and other well-known knot invariants.

Magnetic Forces in an Isopedic Disk

We consider the magnetic forces in electrically conducting thin disks threaded by magnetic fields originating in the external (interstellar) medium. We focus on disks that have dimensionless ratios λ of the mass to flux that are spatially constant, a condition that we term isopedic. For arbitrary distributions of the surface density Σ (which can be nonaxisymmetric and time dependent), we show that the magnetic tension exerts a force in the plane of the disk equal to -1/λ2 times the self-gravitational force. In addition, if the disk maintains magnetostatic equilibrium in the vertical direction, the magnetic pressure, integrated over the z-height of the disk, may be approximated as (1 + η2)/(λ2 + η2) times the gas pressure integrated over z, where η ≡ f||/2πGΣ and f${b f}$ -->|| is the component of the local gravitational field parallel to the plane of the disk. We apply these results to the problem of the stability of magnetized isothermal disks to gravitational fragmentation into subcondensations of a size comparable to the vertical scale height of the disk. Contrary to common belief, such dynamical fragmentation probably does not occur. In particular, the case of the magnetized singular isothermal disk undergoes not dynamical fragmentation into many subcondensations, but inside-out collapse into a single compact object, a self similar problem that is studied in a companion paper (Li & Shu 1997).

Self‐similar Collapse of an Isopedic Isothermal Disk

We study the gravitational collapse of an isothermal disk which is isopedically magnetized (i.e., with a mass-to-flux ratio that is spatially constant). The two theorems concerning magnetic forces in such a disk proven in a companion paper (Shu & Li 1997), plus the self-similar nature of the overall problem, allow a semianalytical treatment. The inflow occurs in an inside-out manner similar to that which applies in the collapse of the unmagnetized singular isothermal sphere (Shu 1977). These two cases (singular sphere and disk) bracket the range of possible collapse behaviors expected for the family of isopedic singular isothermal toroids described by Li & Shu (1996b). Although the strong magnetic fields dilute the effects of self-gravity in the isopedic isothermal disk, they do not prevent its outer parts (envelope) from falling onto the central condensed object (protostar) at a fixed infall rate , even when the field is perfectly frozen to the matter. Indeed, the higher densities supported by the fields in the equilibrium state increase during collapse in comparison with the unmagnetized case. The larger effective speed of sound due to magnetization produces a smaller effect. The flattened geometry enforced by the strong magnetic fields introduces a complication: the appearance of an outwardly propagating shock wave that runs ahead of the region of infall (also studied by Tsai & Hsu 1995 in a different context). We discuss the implications of our results for the magnetic-flux problem and for the formation of centrifugally supported disks in the presence of rotation.

Are there critical points on the boundaries of singular domains?

Generalizing a result of E. Ghys, we prove a general theorem that implies that if a rational functionf of the Riemann sphere of degree ≧2 leaves invariant a singular domainC (a disk or a ring) on which the rotation number off satisfies a diophantine condition, provided that on $$\bar C$$ f is injective, then each boundary component ofC contains critical point off. The injectivity condition is always satisfied for singular disks associated to linearizable periodic elliptic points off(z)=z n +a, withneℕ,n≧2 andaeℂ. We also show that the singular disks, associated to periodic elliptic points off(z)=e az that satisfy a diophantine condition, are unbounded in ℂ. In the end of the paper, we give a survey of the theory of iteration of entire functions of ℂ.

Material and electromagnetic sources of the Kerr-Newman geometry

A method based on the theory of distributions, introduced by the author to analyse Kerr's singularity, is extended to the study of the Kerr-Newman geometry. In this case both the metric tensor and the electromagnetic potential are considered to be distributions defined all over space-time. The second partial derivatives of these quantities, when inserted into the coupled Einstein-Maxwell equations, give distributions with support on the singular ring and equatorial disk. By this means the sources of Kerr-Newman's geometry are determined. They are consistent with a model of a charged material disk made out of a mixture of two surface fluids, gas and dust, rigidly rotating with angular velocity ω=a−1. It is also verified that the material stress-energy tensor, as well as the electromagnetic-current vector, correctly reproduces the known asymptotic values of mass, charge and angular momentum of the collapsed object, provided due account is taken of the energy and angular momentum lying outside the source. Furthermore, a mapping of the electromagnetic field in the vicinity of the singularity is achieved by drawing the electric and magnetic lines of force. A close examination of the boundary conditions obeyed by these lines of force points to the conclusion that the massive equatorial disk is, in fact, a superconductor.

Space–times with distribution valued curvature tensors

A space–time in which in an admissible coordinate system the metric tensor is continuous but has a finite jump in its first and second derivatives across a submanifold will have a curvature tensor containing a Dirac delta function. The support of this distribution may be of three, two, or one dimension or may even consist of a single event. Lichnerowicz’s formalism for dealing with such tensors is modified so as to obtain a formalism in which the Bianchi identities are satisfied in the sense of distributions. The resulting formalism is then applied to the discussion of the Einstein field equations for problems in which the source of the gravitational field is given by a distribution valued stress-energy tensor. Gravitational shocks are also discussed and their theory is compared with that of high-frequency gravitational waves given by Y. Choquet-Bruhat. By considering a class of line sources as obtainable from cylindrical shells by a limiting process, as was proposed by Israel, one may use the distribution formalism developed for hypersurfaces to treat line sources. The line source model proposed by Israel to represent the Kerr metric in the neighborhood of its singular disk is shown to lead to a gravitational mass and angular momentum inconsistent with those of the latter metric. It is proposed to remove this difficulty by changing the assumptions made by Israel concerning the nature of the space–time inside the cylindrical shell which is the support of the distribution in the curvature tensor. The details of the effect of this change are not given in this paper.

Some universally pierced arcs in 𝐸³

A subset X X of E 3 {E^3} is said to be universally pierced if each 2 2 -sphere containing X X can be pierced by a tame arc at each point of X X . We show that an arc A A is universally pierced provided A A has a shrinking point p p such that either p p lies in a tame arc in A A or E 3 − A {E^3} - A has 1 1 -ALG at p p . Applying this result we show the existence of infinitely many wild universally pierced arcs.

A New Class of Knots with Property $P$

It is known that several classes of knots have property P ([1], [3], [5], [9]). In this paper, it will be shown that the special class of knots has property P. To show the above, 3-dimensional homology spheres constructed by Dehn's method will be considered and it will be shown that they are not 3-dimensional homotopy spheres. I am grateful to Prof. R. H. Fox and the referee for helpful comments and suggestions regarding this work. A singular disk in the 3-sphere 5 means a map / of an oriented disk D into 5. For brevity, one may refer to the image D=f(D) as the singular disk. Among the singularities that a singular disk may have perhaps the simplest is a clasping singularity or just clasp. This consists of two mutually disjoint slits S and ,S that are mapped by / topologically onto an arc 5 of D. The singular disks to be considered are those that have only simple clasps. Let us call such a disk an elementary disk. This is a natural class to consider, since it is known that any singular disk in general position can be deformed, without moving the boundary, into an elementary disk [10]. If D is an elementary disk then a regular neighborhood W of D in S is a handlebody, and its boundary dW is an orientable surface of some

More on complements of minimal spanning surfaces

W. R. Alford in volume 91 of the Annals of Mathematics has shown the existence of a knot which has two minimal spanning surfaces whose complements in S are not homeomorphic. The trefoil knot is a companion to the knot. This paper shows that any nontrivial knot k is a companion to a knot K which has at least two minimal spanning surfaces. Introduction. In [1], W. R. Alford exhibited a knot k and two minimal spanning surfaces Sx and S2 for k such that S 3 — S{ are not homeomorphic. The knot was formed by sending the torus T containing the knot I in Fig. 1 faithfully to a regular neighborhood of the trefoil knot. In a later paper [2], Alford and C. B. Schaufele constructed knots with 2 really distinct minimal spanning surfaces; the surfaces do not have homeomorphic complements. The examples were constructed by sending the torus T containing the knot I in Fig. 1 faithfully to a regular neighborhood of the sum of m nice knots. The selection of the knots was strongly influenced by their algebraic properties. The purpose of this paper is to show that any nontrivial knot is a companion to a knot K which has at least two minimal surfaces. The knot K is the image of the knot I in T in Fig. 1 under a faithful homeomorphism of the solid torus T to a regular neighborhood V of the knot I. The Alexander polynomial of K is (2 — t) • (2t — 1) [4] for any nontrivial k used. Thus K had genus at least one. The spanning surfaces for K have genus one, so K has genus one. The surfaces. The surfaces for K are constructed as in [2]. The knot I is spanned by a singular disk in T as shown in Fig. 2. Only one side is shown; the singularities are in heavy lines. Received by the editors June 4,1971 and, in revised form, November 8,1971. AMS (MOS) subject classifications (1970). Primary 55A25; Secondary 57A10.

Some Tameness Conditions Involving Singular Disks

Some Tameness Conditions Involving Singular Disks

Free curves inE3

A 2-sphere in Euclidean 3-dimensional space Ez is called free if it can be pushed into either complementary domain by a map moving no point more than e, for arbitrary e. Such 2-spheres have been the object of much recent attention, although the basic problem of whether they must be tame or not remains unsolved. The purpose of this paper is to take a different direction in this study. We introduce a natural generalization of the term free so that it can be used to describe a A -sphere in En, then direct our attention to free 1-spheres and 2-spheres in E3. Our main tool is Theorem 1, which, roughly speaking, should be viewed as follows: It is well known that if D and E are both polyhedral disks in Ez intersecting only in their interiors (in general position), then E may be altered via a disk replacement process to miss D. Theorem 1 states that even if D were a singular disk in E3, this process would remain valid to an extent.

Some tameness conditions involving singular disks

Some tameness conditions involving singular disks