Homotheticity Research Articles

Projectively induced Kähler cones over regular Sasakian manifolds

Motivated by a conjecture in Loi et al. (Math Zeit 290:599–613, 2018) we prove that the Kähler cone over a regular complete Sasakian manifold is Ricci-flat and projectively induced if and only if it is flat. We also obtain that, up to Da\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal D_a$$\\end{document}—homothetic transformations, Kähler cones over homogeneous compact Sasakian manifolds are projectively induced. As main tool we provide a relation between the Kähler potentials of the transverse Kähler metric and of the cone metric.

Packing Soft Polygons in a Minimum Height Rectangular Target Domain

The paper studies packing polygons of variable shapes, regarding the stretching coefficient, in a rectangular target domain of minimum height. Packing objects of a variable shape have a wide spectrum of applications, e.g, in biology, materials science, mechanics, land allocation, and logistics. Interest in these problems is also due to the modeling of the structures of porous media under pressure, e.g., for creating test models of artificial digital cores. Elements of porous media can be deformed under the influence of an external force, but the mass of each particle remains unchanged. This corresponds to conservation of area for the two-dimensional case. Polygonal objects must be completely contained within the target domain (containment constraint) and do not overlap (non-overlapping constraint), provided they have free translations, continuous rotations, stretch transformations, and conserve their area. The phi-function technique is used for an analytical description of the placement constraints for variable shape polygons. Quasi-phi-functions for describing non-overlapping constraints and phi-functions for describing containment constraints are defined. The packing problem is presented in the form of a nonlinear programming model. A solution strategy is proposed, which consists of the following stages: generation of feasible starting points; search for local minima of the problem of packing soft polygons for each starting point using the decomposition algorithm; choosing the best local minimum found at the previous stage. To search for smart starting arrangements, an optimization algorithm for packing original polygons using their homothetic transformations is applied. Decomposition of the problem of packing polygons of variable shapes is based on an iterative procedure that allows reducing a large-scale problem to a sequence of smaller nonlinear programming problems (linear to the number of objects). Numerical examples are provided for oriented rectangles and non-oriented regular polygons. Keywords: packing, polygons, stretch transformation, rectangular container, quasi-phi-functions, optimization, decomposition algorithm.

A Short Note on a Mus-Cheeger-Gromoll Type Metric

In this paper, we first show that the complete lift $U^{c}$ to $TM$ of a vector field $U$ on $M$ is an infinitesimal fiber-preserving conformal transformation if and only if $U$ is an infinitesimal homothetic transformation of $(M,g)$. Here, $(M, g)$ is a Riemannian manifold and $TM$ is its tangent bundle with a Mus-Cheeger-Gromoll type metric $\tilde{g}$. Secondly, we search for some conditions under which $\left(\overset{h}{\nabla},\tilde{g}\right)$ is a Codazzi pair on $TM$ when $(\nabla, g)$ is a Codazzi pair on $M$ where $\overset{h}{\nabla}$ is the horizontal lift of a linear connection $\nabla$ on $M$. We finally discuss the need for further research.

Closed Form Solution in the Buckling Optimization Problem of Twisted Shafts

The counterpart for Euler’s buckling problem is Greenhill’s problem, which studies the forming of a loop in an elastic beam under torsion. In the context of twisted shafts, the optimal shape of the beam along its axis is searched. A priori form of the cross-section remains unknown. For the solution of the actual problem, the stability equations take into account all possible convex and simply connected shapes of the cross-section. The cross-sections are similar geometric figures related by a homothetic transformation with respect to a homothetic center on the axis of the beam and vary along its axis. The distribution of material along the length of a twisted shaft is optimized so that the beam is of the constant volume and will support the maximal moment without spatial buckling. The applications of the variational method for stability problems are illustrated in this manuscript.

On a simple model for describing convection of the rotating fluid: Integrability, bifurcations and global dynamics

<p style='text-indent:20px;'>The Glukhovsky-Dolzhansky (GD) model arises naturally from geophysical science, which describes rotating fluid convection inside the ellipsoid. This work aims to provide some new insights into the GD model. (<i>i</i>) We first show that, under some conditions there are homothetic transformations which covert the GD model into other similar quadric physical models, therefore, our results on the GD model can be naturally applied to the investigation of these models. (<i>ii</i>) We propose a complete classification of Darboux polynomials and exponent factors for the GD model, which implies that the GD model has no polynomial, rational, or Darboux first integrals. In addition, some integrable cases of the GD model are also given when the physical parameters are allowed to be non-positive. (<i>iii</i>) The existence of global attractor is proved. The stability and local bifurcations of all co-dimension one and two are investigated. Particularly, we show that the GD model undergoes two dynamical transitions as the Rayleigh number increases. (<i>iv</i>) To understand the asymptotic behavior of the orbits for the GD model, we use the Poincaré compactification method to study its dynamical behavior at infinity. More precisely, we prove that the phase portraits of the GD model at infinity consist of an infinite sequence of periodic solutions and two heteroclinic loops. Our results may help us better understand the complex and rich dynamics of rotating fluid convection.</p>

On the homothetic transformations in areal spaces of the submetric class

On the homothetic transformations in areal spaces of the submetric class

The groups of homothetic transformations in areal spaces of the submetric class

The groups of homothetic transformations in areal spaces of the submetric class

Stability optimization for a simultaneously twisted and compressed rod

PurposeThe authors search the optimal distribution of bending flexure along the axis of the rod. For the solution of the actual problem, the stability equations take into account all possible convex, simply connected shapes of the cross-section. The authors study the cross-sections with equal principal moments of inertia. The cross-sections are similar geometric figures related by a homothetic transformation with respect to a homothetic center on the axis of the rod and vary along its axis. The cross-section that delivers the maximum or the minimum for the critical eigenvalue must be determined among all convex, simply connected domains. The optimal form of the cross-section is known to be an equilateral triangle. The distribution of material along the length of a twisted and compressed rod is optimized so that the rod must support the maximal moment without spatial buckling, presuming its volume remains constant among all admissible rods. The static Euler's approach is applicable for simply supported rod (hinged), twisted by the conservative moment and axial compressing force.Design/methodology/approachThe optimization problems for stability of twisted and compressed rods are studied in this manuscript. The complement for Euler's buckling problem is Greenhill's problem, which studies the forming of a loop in an elastic bar under simultaneous torsion and compression (Greenhill, 1883).FindingsFor determining the optimal solution, the authors directly compare the twisted rods with the different lengths and cross-sections using the invariant factors. The solution of optimization problem for simultaneously twisted and compressed rod is stated in closed form.Research limitations/implications(1) The linear stability equations are applied. (2) No nonlinear or postbuckling effects were accounted. (3) The moment-free, ideal boundary conditions on both ends of the rod assumed.Practical implicationsOne of the most common design cases in mechanical engineering is the concurrent compression and twisting of the straight members. The closed-form solution allows the immediate estimation of the optimization effect for axes and rotors in industrial and automotive engineering.Social implicationsThe application of lighter and material-saving structural elements allow the saving fabrication resources, reducing the mass of vehicles and industry machines. The systematic usage of material optimized structural elements assists the stabilization of global energy balance of Earth.Originality/valueAlbeit the governing ordinary differential equations are linear, the application of the optimality conditions leads to the nonlinearity of the final optimization equations. The search of closed form solution of the nonlinear differential equations is one of the mathematically hardest tasks in engineering mathematics. The closed-form solution presents in terms of higher transcendental functions.

Study of the excess property of the L-optimal design for the Laible model

For a rational two-dimensional nonlinear in parameters Laible model used in analytical chemistry, the problem of constructing L-optimal designs is investigated. It is shown that there are two types of optimal designs for this model: saturated (i. e., designs with the number of support points equal to the number of model parameters) and excess (i. e., designs with the number of support points greater than the number of model parameters) and that with some homothetic transformations of the design space, locally L-optimal designs can change the type from saturated to excess and vice versa. An analytical solution to the problem of finding the dependence between the number of the optimal design support points and the values of the model parameters based on the application of a functional approach is proposed. The L-efficiency of D-optimal designs is investigated.

Optimized packing multidimensional hyperspheres: a unified approach.

In this paper an optimized multidimensional hyperspheres packing problem (HPP) is considered for a bounded container. Additional constraints, such as prohibited zones in the container or minimal allowable distances between spheres can also be taken into account. Containers bounded by hyper- (spheres, cylinders, planes) are considered. Placement constraints (non-intersection, containment and distant conditions) are formulated using the phi-function technique. A mathematical model of HPP is constructed and analyzed. In terms of the general typology for cutting & packing problems, two classes of HPP are considered: open dimension problem (ODP) and knapsack problem (KP). Various solution strategies for HPP are considered depending on: a) objective function type, b) problem dimension, c) metric characteristics of hyperspheres (congruence, radii distribution and values), d) container's shape; e) prohibited zones in the container and/or minimal allowable distances. A solution approach is proposed based on multistart strategies, nonlinear programming techniques, greedy and branch-and-bound algorithms, statistical optimization and homothetic transformations, as well as decomposition techniques. A general methodology to solve HPP is suggested. Computational results for benchmark and new instances are presented.

Home Production with Time to Consume

We construct a general equilibrium model with home production where consumers choose how to spend their off-market time using market consumption purchases. The time-intensities and productivities of different home production activities determine the degree to which variation in income and relative market prices affects both the composition of expenditure and market labor hours per worker. When accounting for time to consume, homothetic utility functions can still generate non-linear expansion paths as wages increase. For the United States substitution effects due to relative price changes dominate income effects from wage growth in contributing to the rise in the services share and the fall in hours per worker. Quality improvements to goods and services have roughly kept pace with each other, so that changes to sectoral produc- tion efficiencies are the primary driver of relative price variation.

The consumer's index

AbstractConsumer's surplus measures the area under the demand curve between two prices, but is path dependent. There exists a path such that consumer's surplus tracks utility and an explicit formula is known for CES utilities. This paper shows that the CES‐based formula holds for any homothetic utility, and I call it the consumer's index. The index modifies consumer's surplus in two ways: the change in income is measured by its growth factor and the area under the demand curve is normalized by income.

Sagittal height differences of disposable soft contact lenses.

We aim to improve contact lens fitting by using an innovative and simple photogrammetry imaging system to find the sagittal height (SAG) of soft contact lenses. Eleven different types of commercially available soft contact lenses were measured, and five different lenses per package of each lens type were evaluated. The lenses were inserted in a polymethyl methacrylate cell with parallel faces containing a solution of saline and fluorescein to improve the contrast against the background. For every lens, two photographs from the top view and five photographs from the side view were taken. Using homothetic transformations, we calculated the sagittal height. The SAG of all lenses ranged from 3450 to 3907μm. Differences can be appreciated between SAG of different materials. Performing an ANOVA test, we confirm that the intra-packaging sagitta is reliable for every lens. Comparing the measured SAG with the calculated spherical one, we confirm that the majority of lenses, eight out of eleven, have a spherical geometry for the internal side. Finally, we are able to group the type of lenses that present similar SAG apart from the data reported on the blister. Optical coherence tomography measurement of the eye sagitta over a given chord helped finding the first lens to fit, because matching contact lens sagitta and ocular sagitta is the key for a good fitting. In our work, we found that the majority of brands use single sphere geometry for internal surface of disposable soft contact lenses.

Excess and saturated D-optimal designs for the rational model

For a rational two-dimensional nonlinear in parameters model used in analytical chemistry, we investigate how homothetic transformations of the design space affect the number of support points in the optimal designs. We show that there exist two types of optimal designs: a saturated design (i.e. a design with the number of support points which is equal to the number of parameters) and an excess design (i.e. a design with the number of support points which is greater than the number of parameters). The optimal saturated designs are constructed explicitly. Numerical methods for constructing optimal excess designs are used.

Null Electromagnetic Fields from Dilatation and Rotation Transformations of the Hopfion

The application of topology concepts to Maxwell equations has led to the developing of the whole area of electromagnetic knots. In this paper, we apply some symmetry transformations to a particular electromagnetic knot, the hopfion field, to get a new set of knotted solutions with the properties of being null. The new fields are obtained by a homothetic transformation (dilatation) and a rotation of the hopfion, and we study the constraints that the transformations must fulfill in order to generate valid electromagnetic fields propagating in a vacuum. We make use of the Bateman construction and calculate the four-potentials and the electromagnetic helicities. It is observed that the topology of the field lines does not seem to be conserved as it is for the hopfion.

The limit shapes of midpoint polygons in ℝ3

For a polygon in the [Formula: see text]-dimensional Euclidean space, we give two kinds of normalizations of its [Formula: see text]th midpoint polygon by a homothetic transformation and an affine transformation, respectively. As [Formula: see text] goes to infinity, the normalizations will approach “regular” polygons inscribed in an ellipse and a generalized Lissajous curve, respectively, where the curves may be degenerate. The most interesting case is when [Formula: see text], where polygons with all its [Formula: see text]th midpoint polygons knotted are discovered and discussed. Such polygonal knots can be seen as a discrete version of the Lissajous knots.

Optimal Portfolio Choice Over the Life Cycle with Social Security

This paper examines how households should optimally allocate their portfolio choices between risky stocks and risk-free bonds over their lifetime. Traditional lifecycle models in previous work suggest that the allocation toward stocks should start high (near 100%) early in life and decline over a person’s age as human capital depreciates. These models also suggest that, with homothetic utility, the allocation should be roughly independent of a household’s permanent income. The actual empirical evidence, however, indicates more of a “hump” shape allocation over the lifecycle; the lifetime poor also hold a smaller percentage of their portfolio in stocks relative to higher income groups. Households, therefore, appear to be making considerable “mistakes” in their portfolio allocation. Target date funds, which have grown enormously during the past five years, aim to simplify the investment process in a manner consistent with the predictions of this traditional model. We reconsider the portfolio choice allocation in a computationally-demanding lifecycle model in which households face uninsurable wage shocks, uncertain lifetime as well as a progressive and wage-indexed social security system. Social security benefits, therefore, are correlated with stock returns at a low frequency that is more relevant for lifecycle retirement planning. We show that this model is able to more closely replicate the key stylized facts of portfolio choice. In fact, when calibrated to the age-based income-wealth ratios found in the Survey of Consumer Finances, we demonstrate that the portfolio allocation “mistakes” being made by the vast majority of households actually lead to larger levels of welfare relative to the traditional advice incorporated in target date funds.

Dependence of inertial confinement fusion ignition energy threshold on electron thermal conduction

In inertial confinement fusion, ignition conditions of the thermonuclear deuterium–tritium fusion reactions depend on the electron thermal heat conduction that affects the hot-spot mass accretion and energy balance. One-dimensional hydrodynamic calculations have been performed to simulate the implosion of a capsule directly irradiated by laser beams. For these calculations, the laser-capsule configuration has been scaled using homothetic transformations to scan the transition to ignition while keeping the implosion velocity constant. The electronic heat conduction has been modified by two alternative ways: (i) the classical Spitzer heat flux has been harmonically limited to the free streaming limit using a flux limit factor φ, and (ii) the classical Spitzer coefficient is reduced by a multiplicative factor α < 1. A change of the thermal conductivity affects the performances of the implosion and it can be beneficial or harmful. Parametric studies have been performed for both alternatives, looking for the kinetic energy threshold that generates a unitary energy gain as a function of parameters φ or α, respectively. These studies show how the energy threshold is modified by acting on the heat flux and that there is a minimum ignition energy that can be tuned by appropriately reducing the heat conduction. A simple qualitative model is developed to understand the relation between energy threshold and heat conduction.

On the possibility of singularities on the ambient boundary

The order horismos induces the Zeeman [Formula: see text] topology, which is coarser than the Fine Zeeman Topology [Formula: see text]. The causal curves in a spacetime under [Formula: see text] are piecewise null. [Formula: see text] is considered to be the most physical topology in a spacetime manifold [Formula: see text], as the group of homeomorphisms of [Formula: see text] is isomorphic to the group of homothetic transformations of [Formula: see text]. [Formula: see text] was used in the a ambient boundary-ambient space cosmological model, in order to show that there is no possibility of formation of spacetime singularities. In this paper, we question this result, by reviewing the corresponding papers, and we propose new questions toward the improvement of this model.

Excess of locally D-optimal designs and homothetic transformations

The paper is devoted to the study of homothety’s influence on the number of optimal design support points under fixed values of a regression model’s parameters. The Ayen–Peters two-dimensional nonlinear in parameters model used in analytical chemistry is considered. It is shown that the number of optimal design support points must be greater than or equal to the number of parameters depending on certain conditions. The optimal designs with the minimal number of support points are constructed explicitly. Some numerical methods for constructing designs with greater number of points (we suggest to call them excess designs) are used.