Asymmetric Convex Research Articles

Optimal Partitioning of Probability Distributions under General Convex Loss Functions in Selective Assembly

Selective assembly is an effective approach for improving the quality of a product which is composed of two mating components. This article studies optimal partitioning of the dimensional distributions of the components in selective assembly. It extends previous results for squared error loss function to cover general convex loss functions, including asymmetric convex loss functions. Equations for the optimal partition are derived. Assuming that the density function of the dimensional distribution is log-concave, uniqueness of solutions is established and some properties of the optimal partition are shown. Some numerical results compare the optimal partition with some heuristic partitioning schemes.

First-principles study of phase stability of Gd-doped EuO and EuS

Phase diagrams of isoelectronic Eu${}_{1\ensuremath{-}x}$Gd${}_{x}$O and Eu${}_{1\ensuremath{-}x}$Gd${}_{x}$S quasibinary alloy systems are constructed using first-principles calculations combined with a standard cluster-expansion approach and Monte Carlo simulations. The oxide system has a wide miscibility gap on the Gd-rich side but forms ordered compounds on the Eu-rich side, exhibiting a deep asymmetric convex hull in the formation enthalpy diagram. The sulfide system has no stable compounds. The large difference in the formation enthalpies of the oxide and sulfide compounds is attributed to the contribution of local lattice relaxation, which is sensitive to the anion size. The solubility of Gd in both EuO and EuS is in the range of 10%--20% at room temperature and quickly increases at higher temperatures, indicating that highly doped disordered solid solutions can be produced without the precipitation of secondary phases. We also predict that rocksalt GdO can be stabilized under appropriate experimental conditions.

Querying Approximate Shortest Paths in Anisotropic Regions

We present a data structure for answering approximate shortest path queries in a planar subdivision from a fixed source. Let $\rho\geqslant1$ be a real number. Distances in each face of this subdivision are measured by a possibly asymmetric convex distance function whose unit disk is contained in a concentric unit Euclidean disk and contains a concentric Euclidean disk with radius $1/\rho$. Different convex distance functions may be used for different faces, and obstacles are allowed. Let $\varepsilon$ be any number strictly between 0 and 1. Our data structure returns a $(1+\varepsilon)$ approximation of the shortest path cost from the fixed source to a query destination in $O(\log\frac{\rho n}{\varepsilon})$ time. Afterwards, a $(1+\varepsilon)$-approximate shortest path can be reported in $O(\log n)$ time plus the complexity of the path. The data structure uses $O(\frac{\rho^2n^3}{\varepsilon^2}\log\frac{\rho n}{\varepsilon})$ space and can be built in $O(\frac{\rho^2n^3}{\varepsilon^2}(\log\frac{\rho n}{\varepsilon})^2)$ time. Our time and space bounds do not depend on any other parameter; in particular, they do not depend on any geometric parameter of the subdivision such as the minimum angle.

Irreversible Investments Revisited

A non-linear dynamic model in two state variables, two controls and three cost terms is presented for the purpose of finding the optimal combination of exploitation and capital investment in optimal renewable resource management. Non-malleability of capital is, in other words, incorporated in the model through an asymmetric convex cost-function of investment, and investments can be both positive and negative. Exploitation is controlled through the utilisation rate of available capital. A novel feature in this model is that there are fixed costs associated with the available capital whether it is utilised or not. In contrast to most of the previous literature both state variables enter the objective function.

Enumeration of symmetry classes of convex polyominoes on the honeycomb lattice

We enumerate the symmetry classes of convex polyominoes on the hexagonal (honeycomb) lattice. Here convexity is to be understood as convexity along the three main column directions. We deduce the generating series of free (i.e. up to reflection and rotation) and of asymmetric convex hexagonal polyominoes, according to area and half-perimeter. We give explicit formulas or implicit functional equations for the generating series, which are convenient for computer algebra. Thus, computations can be carried out up to area 70.

Multiple brake orbits on convex hypersurfaces under asymmetric pinch conditions

In this paper, we give an example of an asymmetric convex compact hypersurface possessing precisely n brake orbits. For H ∈ C 1 ( R 2 n , R ) satisfying H ( - p , q ) = H ( p , q ) for all ( p , q ) ∈ R n × R n , we prove multiplicity result of the brake orbits on H - 1 ( 1 ) pinched by not necessarily symmetric hypersurfaces.

Closed Characteristics on Asymmetric Convex Hypersurfaces in R 2n and the Corresponding Pinching Conditions

In this paper, we construct first a new concrete example of asymmetric convex compact C 1,1-hypersurfaces in R 2n possessing precisely n closed characteristics. Then we prove multiplicity results on the closed characteristics on convex compact hypersurfaces in R 2n pinched by not necessarily symmetric convex compact hypersurfaces.

Stability Theorems for Two Measures of Symmetry

This article concerns two measures of central symmetry that characterize cones as the most asymmetric convex bodies. Stability estimates are established that provide information on the deviation of a convex body from a cone if the corresponding measure of symmetry is close to its maximum value.

An Investigation of Techniques for Asymmetry Rectification

This paper is concerned with symmetry and asymmetry in objects. Earlier studies on the subject have dealt with the detection of symmetry. We study the problem of asymmetry rectification, i.e., given an asymmetric profile, suggest geometric changes to make it symmetric. We have investigated two techniques for asymmetry rectification. This paper reports on the techniques and the results. First, we analyze the problem using skeleton representations and propose a simple algorithm for rectifying asymmetric convex profiles. Next, we outline an optimization scheme for the rectification of convex and nonconvex profiles, in 2D and 3D. Implemented examples are presented for both algorithms.

Convex combinations of unitary operators in von Neumann algebras

The unitary rank of an element T in a von Neumann algebra U is defined as the smallest number u ( T ) for which there is a convex combination of unitaries from U of length u ( T ) and equalling T . We determine the unitary rank for every element T in the closed unit ball of U in terms of the index i ( T ) of T and the distance α ( T ) from T to the group of invertible elements in U . If i ( T ) = 0, then u ( T ) ⩽ 2; if i ( T ) ≠ 0, then u ( T ) = n when n − 1 < 2(1 − α ( T )) −1 ⩽ n , and u ( T ) = ∞ when α ( T ) = 1. We also determine precisely which asymmetric convex decompositions of T can be realized. We show that only an approximate version of these results holds in a general C ∗ -algebra.

Plane polygons and a conjecture of Blaschke’s

The Radon transform of a plane domain is a random variable assigning to each line in the plane the chord length of its intersection with the domain. The probability distribution of this random variable does not characterize the domain, but it is shown to characterize a sufficiently asymmetric convex polygon. Under weaker assumptions, a convex polygon is characterized by this distribution, up to a finite number of rearrangements.

Plane polygons and a conjecture of Blaschke’s

The Radon transform of a plane domain is a random variable assigning to each line in the plane the chord length of its intersection with the domain. The probability distribution of this random variable does not characterize the domain, but it is shown to characterize a sufficiently asymmetric convex polygon. Under weaker assumptions, a convex polygon is characterized by this distribution, up to a finite number of rearrangements.

Compounds of asymmetric convex bodies

Mahler obtained inequalities relating an0-symmetric convex body with its compounds. These results are extended to asymmetric convex bodies.

On a Measure of Asymmetry of Convex Bodies

In the present note we discuss some properties of a ‘measure of asymmetry’ of convex bodies in n-dimensional Euclidean space. Various measures of asymmetry have been treated in the literature (see, for example, (1), (6); references to most of the relevant results may be found in (4)). The measure introduced here has the somewhat surprising property that for n ≥ 3 the n-simplex is not the most asymmetric convex body in En. It seems to be the only measure of asymmetry for which this fact is known.