Coefficients Of Inequalities Research Articles

Improved upper bounds for the information rates of the secret sharing schemes induced by the Vámos matroid

An access structure specifying the qualified sets of a secret sharing scheme must have information rate less than or equal to one. The Vámos matroid induces two non-isomorphic access structures V 1 and V 6 , which were shown by Martí-Farré and Padró to have information rates of at least 3/4. Beimel, Livne, and Padró showed that the information rates of V 1 and V 6 are bounded above by 10/11 and 9/10 respectively. Here we improve those upper bounds to 8/9 for V 1 and 17/19 for V 6 . We also indicate a general method that allows one to read off an upper bound for the information rate of V 6 directly from the coefficients of any non-Shannon inequality with certain properties, properties that hold for all 4-variable non-Shannon inequalities known to the author.

The pricing of perpetual convertible bond with credit risk

Convertible bond gives holder the right to choose a conversion strategy to maximize the bond value, and issuer also has the right to minimize the bond value in order to maximize equity value. When there is default occurring, conversion and calling strategies are invalid. In the framework of reduced form model, we reduce the price of convertible bond to variational inequalities, and the coefficients of variational inequalities are unbounded at the original point. Then the existence and uniqueness of variational inequality are proven. Finally, we prove that the conversion area, the calling area and the holding area are connected subsets of the state space.

State Income Inequality and Presidential Election Turnout and Outcomes*

Objective. This study examines the links among income inequality, voter turnout, and electoral choice at the state level in recent presidential elections.Methods. We introduce two new state‐level ecological data sets, estimated annual Gini coefficients of income inequality from 1969 to 2004 and a measure of income segregation across Census tracts within states in 1999. We test for associations among inequality, turnout, and party preference with cross‐sectional, fixed‐effects, and multilevel analyses.Results. The cross‐sectional effect of inequality on voter turnout and electoral choice is ambiguous. However, a fixed‐effects analysis links higher income inequality to lower voter turnout and also to a stronger Democratic vote. Multilevel results indicate that higher levels of economic segregation likewise are associated with depressed turnout, after controlling for individual voter characteristics and for state‐level income.

Does trade and technology transmission facilitate convergence? The role of technology adoption in reducing the inequality of nations

Based on stylized evidence showing variation of the Gini coefficients of income inequality across skill cohorts with the rapid rise in trade in technology‐intensive goods, the transmission effects of technology diffusion and income inequality are explored in a global Computable General Equilibrium (CGE) framework. An exogenous technology shock transmitted via trade from the United States induces productivity growth in developing regions. This spillover in technology – aided by absorptive capability, better governance and institutions, technological symmetry and social acceptance – causes income to increase and income inequality to decline. The transmission of technology facilitates convergence of inequality between nations.

Income inequality and population health: Correlation and causality

A large literature now exists on the cross-national correlation between income inequality and population health, but existing studies suffer from sparse data, poor operationalization of income inequality, and the use of low-power statistical models. This paper sets out to estimate the ecological correlation between income inequality and indicators of population health in a very broad panel of countries, to demonstrate that this relationship is largely non-artifactual, and to test whether this relationship might be causal. Gini coefficients of national income inequality in 1970 and 1995 are correlated with life expectancy, infant mortality rates, and murder rates, controlling for national income per capita. In cross-sectional analyses, inequality is significantly correlated with life expectancy, infant mortality, and (inconsistently) the murder rate. The health correlations are shown to be not primarily due to the “convexity effect” of the non-linear relationship between individual income and individual health, which seems to account for no more than one-third of the relationship between inequality and health, and likely much less. Change in inequality 1970–1995 is significantly related to change in life expectancy and infant mortality, suggesting a causal relationship, but these correlations are not robust with respect to sample or controls. It can be concluded that there is a strong, consistent, statistically significant, non-artifactual correlation between national income inequality and population health, but though there is some evidence that this relationship is causal, the relative stability of income inequality over time in most countries makes causality difficult to test.

Pointwise Error Estimates for Scalar Conservation Laws with Piecewise Smooth Solutions

We introduce a new approach to obtain sharp pointwise error estimates for viscosity approximation (and, in fact, more general approximations) to scalar conservation laws with piecewise smooth solutions. To this end, we derive a transport inequality for an appropriately weighted error function. The key ingredient in our approach is a one-sided interpolation inequality between classical L1 error estimates and Lip+ stability bounds. The one-sided interpolation, interesting for its own sake, enables us to convert a global L1 result into a (nonoptimal) local estimate. This, in turn, provides the necessary bounds on the coefficients of the above-mentioned transport inequality. Estimates on the weighted error then follow from the maximum principle, and a bootstrap argument yields optimal pointwise error bound for the viscosity approximation. Unlike previous works in this direction, our method can deal with finitely many waves with possible collisions. Moreover, in our approach one does not follow the characteristics but instead makes use of the energy method, and hence this approach could be extended to other types of approximate solutions.

Nonadditive Set Functions on a Finite Set and Linear Inequalities

A set function is a function whose domain is the power set of a set, which is assumed to be finite in this paper. We treat a possibly nonadditive set function, i.e., a set function which does not satisfy necessarily additivity, ϕ(A)+ϕ(B)=ϕ(A∪B) forA∩B=∅, as an element of the linear space on the power set. Then some of the famous classes of set functions are polyhedral in that linear space, i.e., expressed by a finite number of linear inequalities. We specify the sets of the coefficients of the linear inequalities for some classes of set functions. Then we consider the following three problems: (a) the domain extension problem for nonadditive set functions, (b) the sandwich problem for nonadditive set functions, and (c) the representation problem of a binary relation by a nonadditive set function, i.e., the problem of nonadditive comparative probabilities.

Plant size and spatial pattern in a natural population of Myosotis micrantha

Abstract. We examined spatial distributions and plant sizes along a transect through a natural population of a winter annual, Myosotis micrantha. A size hierarchy existed, as indicated by high values of Gini coefficients of inequality for plant mass and correlated measures. Plants with no immediate conspecific neighbors were larger than plants with one or more near neighbors, suggesting that competition from near neighbors depressed plant size. However, there was strong positive spatial autocorrelation in plant size: large plants were associated with large neighbors and small ones with small neighbors. Plant size was also positively correlated with the combined biomass of near neighbors. The population formed a two‐phase mosaic of patches of relatively large plants alternating with patches of smaller plants. The data suggest that individual plants compete with conspecifics, but the effects of competition are symmetrical. The most likely explanations for this spatially structured size hierarchy are variation in plant density, patchy distribution of resources, or a combination of the two.

Lifted cover facets of the 0–1 knapsack polytope with GUB constraints

Facet-defining inequalities lifted from minimal covers are used as strong cutting planes in algorithms for solving 0–1 integer programming problems. In this paper we extend the result of Balas and Zemel by giving a set of inequalities that determines the lifting coefficients of facet-defining inequalities of the 0–1 knapsack polytope for any ordering of the variables to be lifted. We further generalize the result to obtain facet-defining inequalities for the 0–1 knapsack problem with generalized upper bound constraints.

The anti-join composition and polyhedra

Given a clutter L , we associate with it the covering polyhedron Q L , that is the dominant of the convex hull of incidence vectors of all the covers of L . In this paper we describe a binary composition operation, the anti-join, which combines a pair of clutters L 1 and L 2 to give a new clutter L . The anti-join operation is, in some sense, the “dual” of the join operation, introduced by Cunningham [10]. In fact, it has the property that the blocker of the clutter L obtained by joining two clutters L 1 and L 2, is the anti-join of the blockers of L 1 and L 2. For such an operation we show how the linear descriptions of the polyhedra Q( L 1) and Q( L 2) have to be combined to produce a linear description of the polyhedron Q L . Moreover, given a set F ⊆ {λ: 0⩽λ⩽1} such that {0, 1} ⊆ F and aϵ F if and only if (1−a)ϵ F , we define the F -property for covering polyhedra as a proper generalization of the Fulkerson property, to which it reduces for F={0, 1} . We prove that the anti-join operation preserves the F -property. This implies the characterization of the coefficients of the facet-defining inequalities for the cycle and cocycle polyhedra associated with graphs noncontractible to the four-wheel W 4.

On tightening cover induced inequalities

Here we describe computationally efficient procedures for tightening cover induced inequalities by using 0–1 knapsack constraints and, if available, cliques whose variables are included in the cover. An interesting application is the case where the cover is implied by the knapsack constraint. The tightening is achieved by increasing the coefficients of the cover inequality. The new constraint is 0–1 equivalent to and LP tighter than the original one. The computational complexity of the procedures is O( n log n), where n is the number of variables in the cover.

Some comparison and oscillation results for first-order differential equations and inequalities with a deviating argument

In this paper we are dealing with first-order differential equations and inequalities with a deviating argument and we give some new necessary, sufficient, and necessary and sufficient conditions concerning: (a) the comparison of oscillatory and asymptotic behavior of solutions of: (1) first-order vector differential equations and, corresponding to them, operator and scalar differential equations with a deviating argument and (2) first-order differential equations and, corresponding to them, inequalities with a deviating argument; and (b) the existence of a certain type of nonoscillatory solutions of differential equations and inequalities with a deviating argument. What is important, in our opinion, is the fact that some of the obtained results concern not only the cases of constant sign coefficients, but also the cases where the coefficients of equations and inequalities under consideration are of non-constant sign.

Income Inequality in some Western Countries: Patterns and Trends

This article is concerned with pre‐tax total income distribution in 18 “western” countries. Its objective is a tentative comparison among national income distributions and the study of trends since the 1950s. Similar issues have recently been analysed in a number of studies. This analysis differs from them in three main respects. Firstly, it refers to a group of nations which excludes developing countries. In spite of substantial differences in political and socio‐economic structures in this group of countries, processes of income distribution have a great degree of similarity. Income statistics are then more “homogeneous” and their comparison more meaningful than in studies covering developed and developing nations. Secondly, the article differs in the methodology it uses. Other studies have relied on “strict comparisons” of income statistics, for example they have compared values of coefficients of inequality and of income shares. In this article “loose classification”, for example categorisation of countries into groups (see following section), is used as a means of comparing income distributions. This makes it possible to allow, within limits, for the fact that the value of the data can be different from the one which is observed, even when statistics seem “fairly” accurate and comparable. Thirdly, the possible influence of the data source on each country's position is examined by considering a plurality of sources whenever possible. Intentionally, this article does not include discussion of most of the problems which exist in the field of income distribution—including those concerning coverage and reliability of data. It is expected that the reader keeps them and their various caveats well in mind when evaluating the empirical evidence which is presented. The explanation of differences in income distribution structures is outside the scope of this article.

An Iterative Method of General Planetary Theory

This paper deals with an iterative version of the general planetary theory. Just as in Airy's Lunar method the series in powers of planetary masses are replaced here by the iterations to achieve improved approximations for the coefficients of planetary inequalities. The right-hand members of the equations of motion are calculated in closed formulas, and no expansion in powers of small corrections to the planetary coordinates is needed. For the N-planet case this method requires the performance of the analytical operations on a computer with power series of 4N polynomial variables, the coefficients being the exponential series of N-1 angular arguments. To obtain numerical series of planetary motion one has to solve the secular system using Birkhoff's normalization or the Taylor series in powers of time. A slight modification of the method in the resonant case makes it valid for the treatment of the main problem of the Galilean satellites of Jupiter.

Faces of the Gomory polyhedron for cyclic groups

A two parameter method is given for generating faces of Gomory's master polyhedron for cyclic groups. The paper shows how to specify coefficients of the linear inequality defining a face directly without requiring a special group minimization algorithm. Included are results which specify a unique representation of a cyclic group in terms of restricted multiples of an appropriately selected pair of elements. A computer study indicates that the two parameter method is very efficient, requiring roughly the same amount of computation as Gomory's “θ-method,” but capable of generating a substantially larger number of faces.

LXVIII. On a method of calculating the coefficients of the lunar inequalities

LXVIII. <i>On a method of calculating the coefficients of the lunar inequalities</i>

VIII. On the theory of the moon

When I commenced the investigations relating to the theory of the moon which I have had the honour to communicate to the Society, I proposed to show how, by a different but more direct method, the numerical results given by M. Damoiseau might be obtained. The approximations were in fact carried much further by M. Damoiseau than had been done before, and the details which accompany M. Damoiseau's work evince at once the immense labour of the undertaking, and inspire confidence in the accuracy of the results offered. But the state of the question is now changed by the appearance of M. Plana’s admirable work, entitled “Théorie du Mouvement de la Lune,” in which, although M. Plana's employs the same differential equations as those used by M. Damoiseau, and obtains in the same manner finally the expressions for the coordinates of the moon, in terms of the mean longitude by the reversion of series, yet M. Plana’s expressions have a very different analytical character and im­portance, from the circumstance that the author develops all the quantities intro­duced by integration, according to powers of the quantity called m , which expresses the ratio of the sun’s mean motion to that of the moon. In this form of the expression the coefficients of the different powers of m , of the eccentricity, &amp;c., are determinate, as are, for example, the numerical coefficients in the expression for the sine in terms of the arc, and other similar series. An inestimable advantage results from this pro­cedure, which more than compensates for the great increase of labour it occasions, by diminishing the danger of neglecting any terms of the same order as those taken into account, and by affording the means of verifying many terms long before final and complete results shall have been obtained independently by myself or any other person. By treating the differential equations in which the time is the independent variable, as I have proposed, similar results to those of M. Plana may be obtained directly; but the calculations which are required in either method are so prodigiously irksome and laborious, that until identical expressions have actually been obtained independently, to the extent of every sensible term, the theory of the moon cannot, I think, be considered complete. It might, indeed, be supposed that already, through the labours of mathematicians, from Clairaut to the present time, the numerical values of the coefficients of the different inequalities were ascertained with sufficient accuracy for practical purposes, and that any further researches connected with the subject would be more likely to gratify curiosity than to lead to any useful result. Astronomical observations are now made with so great precision, that the numerical values of the coefficients are wanted to at least the tenth of a second of space: very few, however, of the coefficients of MM. Damoiseau and Plana agree so nearly, and some differ much more, as may be seen in the following comparison of the numerical values of the coefficients of some of the arguments in the expression for the true longitude of the moon in terms of her mean longitude, being indeed those which differ the most.