Terms Of Cones Research Articles

Spinning Cones as Pumps, Degassers and Level Controllers in Mechanically Stirred Tanks

The literature on spinning cones as pumps and degassers is reviewed. Experiments on a spinning cone rig designed to measure spinning cone-pumping rates are described. A large number of experiments were carried out, measuring pumping rates as a function of cone angle and immersion depth. Cone half angles were from 15 to 60°. Most of the tests were done with water but a number of runs were carried out with 75% by volume glycerol solution. This changed the viscosity and fluid density. An equation is proposed to predict the volumetric pumping rate (Q) of a cone in terms of its geometry and the physical properties of the fluids. Experiments with gassed fluids indicate that the liquid pumping rate of a spinning cone is independent of any gas present. It was also found that both outside and inside surfaces of the cone contribute equally to the fluid pumping process. The effectiveness of spinning cones as a degasser and/or level controller have been tested in mechanically agitated baffled vessels using cones mounted on the agitator shaft at the liquid surface. Under gassed conditions and with surfactants added to the liquid, the spinning cone was shown to be effective in controlling the level and reducing the gas voidage over a wide range of operating conditions. The effectiveness of the cone as a defoamer appears to scale at equal tip speed, suggesting that shear rate is the dominant mechanism. Tests were done on 2 ft (0.61 m) and 3 ft (0.914 m) diameter stirred tanks.

The Γ-algorithm and some applications

In this paper the power of the Γ-algorithm for obtaining the dual of a given cone and some of its multiple applications is discussed. The meaning of each sequential tableau appearing during the process is interpreted. It is shown that each tableau contains the generators of the dual cone of a given cone and that the algorithm updates the dual cone when new generators are incorporated. This algorithm, which is based on the duality concept, allows one to solve many problems in linear algebra, such as determining whether or not a vector belongs to a cone, obtaining the minimal representations of a cone in terms of a linear space and an acute cone, obtaining the intersection of two cones, discussing the compatibility of linear systems of inequalities, solving systems of linear inequalities, etc. The applications are illustrated with examples.

A GENERAL FRAMEWORK FOR CONVEX RELAXATION OF POLYNOMIAL OPTIMIZATION PROBLEMS OVER CONES

The class of POPs (Polynomial Optimization Problems) over cones covers a wide range of optimization problems such as 0-1 integer linear and quadratic programs, nonconvex quadratic programs and bilinear matrix inequalities. This paper presents a new framework for convex relaxation of POPs over cones in terms of linear optimization problems over cones. It provides a unified treatment of many existing convex relaxation methods based on the lift-and-project linear programming procedure, the reformulation- linearization technique and the semidefinite programming relaxation for a variety of problems. It also extends the theory of convex relaxation methods, and thereby brings flexibility and richness in practical use of the theory.

An analytic representation for the potential of a uniform elliptic cone

In view of the fact that various gravitating and charged bodies may have singular points on their surface, the problem of the effect of these singularities in the mass distribution on the behavior of potential arises. The analytic representation for gravitational and electrostatic potentials of a uniform elliptic cone in terms of ordinary integrals is derived. Generally, these integrals are somewhat more complicated than the elliptic ones; however, they reduce to elementary functions in the case of a circular cone. The arbitrariness in the way of truncating a cone manifests itself in the possibility of adding various harmonic polynomials. However, the singularity of potential in the vicinity of the vertex of a cone is of an objective and unique character.

A fixed-point index and existence theorems for semilinear equations in cones

The purpose of this thesis is to develop fixed point indices for A-proper semilinear operators defined on cones in Banach spaces and use the results to obtain existence theorems to semilinear equations. We consider semilinear equations of the form Lx = Nx where L is a linear Fredholm operation of index zero, N a nonlinear operator such that L - N is A-proper at zero relative to a projection scheme L. Chapter 1 is an introduction to basic concepts used throughout the thesis, including; Banach spaces, linear operators, A-proper maps, Fredholm operators of index zero, and the definition and properties of the generalised degree for A-proper maps. In Chapter 2, we define a fixed point index for A-proper maps on cones in terms of the generalised degree and derive the basic properties of this index. We then extend the definition to include unbounded sets. A more general fixed point index than that of Chapter 2 is developed in Chapter 3 for A-proper maps based on limits of a finite dimensionally defined index. Properties of the index are given and a definition for unbounded sets is provided. Chapter 4 extends the Lan-Webb fixed point index for weakly inward A-proper at 0 maps to semilinear operators. This index is also extended to include unbounded sets. Existence theorems of positive and non-negative solutions to semilinear equations on cones are established in Chapter 5 using the fixed point indices of Chapters 2, 3, and 4. Finally, in Chapter 6, we apply some of the existence theorems of Chapter 5 to several differential and integral equations. We prove the existence of: a positive solution to a Picard boundary value problem; a non-negative solution to a periodic boundary value problem; and, a non-negative solution to a Volterra integral equation.

Scatterhoarding and Larderhoarding by Red Squirrels: Size, Dispersion, and Allocation of Hoards

We quantified the degree of scatterhoarding and larderhoarding exhibited during late summer and early autumn by 10 individuals within a population of red squirrels ( Tamiasciurus hudsonicus ) in the Cypress Hills, Alberta, Canada. All squirrels made extensive and obvious larderhoards in central middens. In addition, sampling quadrats and behavioral observations revealed that ca. 43% of total stores were scatterhoarded throughout territories. Across individuals, the number of cones scatterhoarded was related inversely to the number stored in the midden. Nearest-neighbor analyses indicated that cones were distributed in a spatially clumped manner throughout each territory. However, the hoards themselves, independent of size, tended to be dispersed randomly. Cones hoarded in middens did not differ from scatterhoarded cones in terms of content of seeds, nor were they treated differently by the squirrels in terms of carrying distance. At the end of our study, seasonal hoarding was not yet complete, but squirrels already had hoarded sufficient cones to meet their energetic requirements for ≥3 months. We conclude that the characteristic pattern of storage of food by red squirrels is a deliberate mixture of distinct larderhoarding and scatterhoarding tactics.

Hyperbolic Polynomials and Interior Point Methods for Convex Programming

Hyperbolic polynomials have their origins in partial differential equations. We show in this paper that they have applications in interior point methods for convex programming. Each homogeneous hyperbolic polynomial p has an associated open and convex cone called its hyperbolicity cone. We give an explicit representation of this cone in terms of polynomial inequalities. The function F(x) = −log p(x) is a logarithmically homogeneous self-concordant barrier function for the hyperbolicity cone with barrier parameter equal to the degree of p. The function F(x) possesses striking additional properties that are useful in designing long-step interior point methods. For example, we show that the long-step primal potential reduction methods of Nesterov and Todd and the surface-following methods of Nesterov and Nemirovskii extend to hyperbolic barrier functions. We also show that there exists a hyperbolic barrier function on every homogeneous cone.

A characterization of paratingent cone andP-subderivative with applications in nonsmooth analysis

We give a new characterization of the paratingent cone in terms of contingent cones, i.e., the paratingent cone to any open set at a boundary point is the upper limit of the contingent cones at the neighboring points. We use this result to characterize the strict differentiability in terms of the contingent directional derivatives. We also define aP-subderivative for continuous functions and develop a subdifferential calculus with applications to optimality conditions in mathematical programming.

Positive semidefinite matrices with a given sparsity pattern

This paper concerns the cone of positive semidefinite matrices which have zeros in prescribed entries. One of the main purposes is to obtain information and, if possible, describe the ranks of external matrices in such cones in terms of the pattern of prescribed zeros. In particular, we study those patterns for which the ranks of extremal matrices are less than a given number.

Semicontinuity of multifunctions connected with optimization with respect to cones

AbstractThis paper studies topological upper and lower semicontinuity of the minimal value multifunction and the solution multifunction for optimization problems, which are defined in terms of cones, subject to perturbations in constraints. It extends the results of Tanino and Sawaragi to finite dimensions and one of Berge to multiple objective optimization problems.

Generalized gradients and paths of descent

There have been several algorithms in the literature that use Clabke's generalized gradients to find descent directions for minimizing Lipschitz functions, The result used to find these directions states that if zero is not a generalized gradient to a Lipschitz function f at xthen there is an y that is a direction of descent for f at all points in a neighborhood of x. In this paper it is shown that, even though there may not be descent directions, there are paths of descent if f is lower semicontinuous. This is done using a characterization of Clabke'S tangent cone in terms of Lipschitz paths.

Nonsmooth analysis of vector-valued mappings with contributions to nondifferentiable programming

We define order Lipschitz mappings from a Banach space to an order complete vector lattice and present a nonsmooth analysis for such functions. In particular, we establish properties of a generalized directional derivative and gradient and derive results concerning a calculus of generalized gradients (i.e., calculation of the generalized gradient of f when f = f1 + f2, f = f · 2, etc.). We show the relevance of the above analysis to nondifferentiaile programming by deriving optimality conditions for problems of the form min f(x) subject to x [euro] S. For S arbitrary we state the results in terms of cones of displacement of the feasible region at the optimal point; when S ={x ∊ A|g(x) ∊ B}, we obtain Kuhn-Tucker type results.

On optimality conditions for maximizations with respect to cones

For a Pareto maximization problem defined in infinite dimensions in terms of cones, relationships among several types of maximal elements are noted and optimality conditions are developed in terms of tangent cones.

Mechanisms of liquid metal ion source operation

A justification for the Taylor cone shape of a liquid metal ion source (LMIS) is given for the case of nonideal electrodes. Arguments are presented for the formation of a jet at the end of the Taylor cone in terms of hydrodynamic and flow considerations rather than in terms of the need to overcome the excessive space charge at the end of very small diameter protuberances at the apex of Taylor cones.

An existence result for maximizations with respect to cones

A sufficient condition is given for the existence of a solution to a generalized Pareto maximization problem in which maximization is defined in terms of cones. This result generalizes the fact that an upper semicontinuous real-valued function achieves its maximum on a compact set.

Wave development on a thinning liquid film

Experimental results are presented for the growth of surface waves on a liquid film that thins as it flows under gravity over the surface of an upright circular cone. The characteristics of the mean film are calculated on the assumption of quasi-parallel flow, and the actual mean thickness found to relate very closely to that found on this basis. The development of the film was found to fall into three phases: the entry zone in which the velocity profile of the film becomes established where no waves are visible, a region of wave growth in which amplitude, wave speed, and wave length all grow, and a final region in which amplitude and wave speed decline as the film thins further although wave length continues to grow. An empirical relationship is presented which expresses the wave number at any point on the cone in terms of the flow rate and a parameter based on the local Reynolds and Weber numbers and cone angle. It was found that for a given flow rate the maximum wave amplitude was reached at a value of wave number of 0·048.

High-speed water jets from vertically accelerated rotating cones

Water jets are produced by vertically accelerating a rotating cone partially filled with water. It is shown that the acceleration of the parabolic meniscus results in a motion similar to that observed in a shaped explosive charge (Monroe jet). Acceleration of the cone is effected by means of an inductive electromagnetic accelerating device (conical pinch) whose theory is developed in terms of the WKB approximation. A second-order inviscid theory for the motion of the fluid in the cone in terms of the Penney-Price linearization procedure is presented and it is shown that good agreement for the jet head velocity can be achieved for low velocities. At higher velocities, experimental results appear to lag behind the theoretical ones, probably owing to the dispersal of the jet head through viscous drag with the surrounding atmosphere. The shape of the jet at early times is well represented by first-order theory.

Manifestly Conformal-Covariant Expansion on the Light Cone

The isomorphism of the conformal algebra on space-time to the orthogonal O(4,2) algebra is exploited to derive in a manifestly covariant way an operator-product expansion on the light cone in terms of irreducible operator representations of the conformal algebra. The expansion provides for a solution of the causality problem for operator expansion on the light cone. Additional properties of the conformally covariant expansion, as well as its relation to the conformally invariant three-point function, are discussed.

A Representation of the Solutions of the Darboux Equation in Odd-Dimensional Spaces

It is shown that determining a function from its averages over all spheres passing through the origin leads to an explicit representation of the even solutions of the Darboux equation in the exterior of the characteristic cones in terms of the hyperboloidal means of the boundary data on the cones.

Singularities of current commutators on the light cone

We consider the forward proton matrix element of the electromagnetic current commutator and express its singularities on the light cone in terms of the scaling functions. This relation between the light cone and the deep inelastic region implies a family of sum rules for the electroproduction structure functions and connects the equal time commutators with the scaling functions. In particular the operator part of the Schwinger term and the related divergences of the electromagnetic mass of the proton are expressed in terms of measurable quantities.