Distance Inequality Research Articles

Unification of distance inequalities for linear variational problems

In this work a unifying approach is presented that leads to bounds for the distance in natural norms between solutions belonging to different spaces, of well-posed linear variational problems with the same input data. This is done in a general hilbertian framework, and in this sense, well-known inequalities such as Cea’s or Babuska’s for coercive and non-coercive problems are extended and/or refined, as mere by-products of this unified setting. More particularly such an approach gives rise to both an improvement and a generalization to the weakly coercive case, of second Strang’s inequality for abstract coercive problems. Additionally several aspects specific to linear variational problems are the subject of a thorough analysis beforehand, which also allows for clarifications and further refinements about the concept of weak coercivity.

The convex distance inequality for dependent random variables, with applications to the stochastic travelling salesman and other problems

We prove concentration inequalities for general functions of weakly dependent random variables satisfying the Dobrushin condition. In particular, we show Talagrand's convex distance inequality for this type of dependence. We apply our bounds to a version of the stochastic salesman problem, the Steiner tree problem, the total magnetisation of the Curie-Weiss model with external field, and exponential random graph models. Our proof uses the exchangeable pair method for proving concentration inequalities introduced by Chatterjee (2005). Another key ingredient of the proof is a subclass of $(a,b)$-self-bounding functions, introduced by Boucheron, Lugosi and Massart (2009).

Modeling Flexible Pharmacophores with Distance Geometry, Scoring, and Bound Stretching

The study of pharmacophores, i.e., of common features between different ligands, is important for the quantitative identification of "compatible" enzymes and binding species. A pharmacophore-based technique is developed that combines multiple conformations with a distance geometry method to create flexible pharmacophore representations. It uses a set of low-energy conformations combined with a new process we call bound stretching to create sets of distance bounds, which contain all or most of the low-energy conformations. The bounds can be obtained using the exact distances between pairs of atoms from the different low-energy conformations. To avoid missing conformations, we can take advantage of the triangle distance inequality between sets of three points to logically expand a set of upper and lower distance bounds (bound stretching). The flexible pharmacophore can be found using a 3-D maximal common subgraph method, which uses the overlap of distance bounds to determine the overlapping structure. A scoring routine is implemented to select the substructures with the largest overlap because there will typically be many overlaps with the maximum number of overlapping bounds. A case study is presented in which 3-D flexible pharmacophores are generated and used to eliminate potential binding species identified by a 2-D pharmacophore method. A second case study creates flexible pharmacophores from a set of thrombin ligands. These are used to compare the new method with existing pharmacophore identification software.

Binary Almost-Perfect Sequence Sets

Sequence set with lower correlation values is highly desired for engineering applications. However, theoretical results (e.g., Welch bound) show that θ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</sub> ≥ √(N) in general, that is, the maximum out-of-phase autocorrelation and cross-correlation magnitudes of a sequence set is not less than the square root of the sequence period. In this paper, we propose a new concept, namely almost perfect sequence set (APSS), which has the property θ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</sub> ≤ c except for at most m shifts, where c and m are predefined small integers. A uniform method is presented to construct APSS and then the properties of such APSS are discussed. Moreover, a distance inequality on the APSS with m = 1 is obtained and several APSS families such as (2p, 8p + 2, 6, 4) -APSS and (3p, (64p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> +8)/3,9,9) -APSS for any prime p ≥ 5 are constructed based on Paley and Paley partial sequences. Finally, it shows that the APSS can be used to construct LCZ sequences and the properties of such LCZ sequences are presented.

On concentration of self-bounding functions

We prove some new concentration inequalities for self-bounding functions using the entropy method. As an application, we recover Talagrand's convex distance inequality. The new Bernstein-like inequalities for self-bounding functions are derived thanks to a careful analysis of the so-called Herbst argument. The latter involves comparison results between solutions of differential inequalities that may be interesting in their own right.

Fast exhaustive multi-resolution search algorithm based on clustering for efficient image retrieval

To find the best match for a query according to a certain similarity measure, this paper presents a fast exhaustive multi-resolution search algorithm based on image database clustering. Prior to search process, the whole image dataset is partitioned into a pre-defined number of clusters having similar feature contents. For a given query, the proposed algorithm first checks the lower bound of distances in each cluster, eliminating disqualified clusters. Next, it only examines the candidates in the surviving clusters through feature matching. To alleviate unnecessary feature-matching operations in the search procedure, the distance inequality property based on a multi-resolution data structure is employed. Simulation results show that the proposed algorithm guarantees very rapid exhaustive search.

Probability distance inequalities on Riemannian manifolds and path spaces

We construct Otto–Villani's coupling for general reversible diffusion processes on a Riemannian manifold. As an application, some new estimates are obtained for Wasserstein distances by using a Sobolev–Poincaré type inequality introduced by Latała and Oleszkiewicz. The corresponding concentration estimates of the measure are presented. Finally, our main result is applied to obtain the transportation cost inequalities on the path space with respect to both of the L 2-distance and the intrinsic distance. In particular, Talagrand's inequality holds on the path space over a compact manifold.

Node-Capacitated Ring Routing

We consider the node-capacitated routing problem in an undirected ring network along with its fractional relaxation, the node-capacitated multicommodity flow problem. For the feasibility problem, Farkas' lemma provides a characterization for general undirected graphs, asserting roughly that there exists such a flow if and only if the so-called distance inequality holds for every choice of distance functions arising from nonnegative node weights. For rings, this (straightforward) result will be improved in two ways. We prove that, independent of the integrality of node capacities, it suffices to require the distance inequality only for distances arising from (0-1-2)-valued node weights, a requirement that will be called the double-cut condition. Moreover, for integer-valued node capacities, the double-cut condition implies the existence of a half-integral multicommodity flow. In this case there is even an integer-valued multicommodity flow that violates each node capacity by at most one. Our approach gives rise to a combinatorial, strongly polynomial algorithm to compute either a violating double-cut or a node-capacitated multicommodity flow. A relation of the problem to its edge-capacitated counterpart will also be explained.

Comparing invariant distances and conformal metrics on Riemann surfaces

We exhibit sharp inequalities comparing the hyperbolic distance (and hyperbolic metric) to distances (and associated metrics) defined via positve harmonic functions as well as bounded harmonic functions. In the simply connected case, all four inequalities are identities. For the non-simply connected case, we determine precisely when equality can hold: for a pair of points in a distance inequality, or a single point in a metric inequality.

Protein Structures from Distance Inequalities

A computer method for folding protein backbones from distance inequalities is presented. It involves an algorithm that uses a novel approach for handling inequalities through the minimization of a continuous energy function. Tests of the folding algorithm have been carried out on a small protein, the 6PTI (bovine pancreatic trypsin inhibitor) with 56 amino acid residues, and on a medium-size protein, the 1TRM (rat trypsin) with 223 amino acid residues. Reconstructions based on a real-valued distance matrix led to folded three-dimensional structures with root-mean-square values of 0·04 Å when compared with the crystallographic data. The obtained root-mean-square measures were of the order of 1 Å, when distance inequalities were used for the reconstruction. Subsequently, the folding approach has been applied to distance inequalities predicted by neural network techniques that use the amino acid sequence as the only input. The inaccuracy in the inequalities predicted by the neural network was the reason for the root-mean-square value of 5·2 Å. An error analysis of the method for reconstruction was performed and showed that no more than 3% inaccurate distance inequalities could be corrected for. Finally, a simple technique for root-mean-square comparisons of different protein structures is discussed.

Distances in cocomparability graphs and their powers

Let ccp denote the class of cocomparability graphs. We characterize ccp by a distance inequality. As a consequence we show: If G ε ccp , then the kth power G k also belongs to ccp , for any k ⩾ 1. As a further result we get that such graphs G k(Gε ccp,k⩾2) have no induced subgraphs isomorphic to K 1, 4 or K 2, 3.

Erdős-Turán inequalities for distance functions on spheres.

Erdős-Turán inequalities for distance functions on spheres.

Efficient characterizations of n-chromatic absolute retracts

An absolute retract is a graph (without loops) for which every isometric embedding into another graph with the same chromatic number is a coretraction. Absolute retracts of graphs can be regarded as the injective objects of a suitable category: the objects are coloured spaces with integer metrics (such as coloured graphs) and the morphisms are non-expansive mappings preserving colours. Hitherto only bipartite absolute retracts were known to admit efficient characterizations. In this paper several characterizations are established by which n-chromatic absolute retracts can be recognized in polynomial time. These procedures either directly test the solvability of certain systems of distance inequalities under colour constraints or recursively operate on neighbourhood lists and dismantle the graph vertex by vertex.

ORAL: All purpose molecular mechanics simulator and energy minimizer

AbstractA molecular mechanics energy minimizer is presented whose main features are the “floating blocks” and “isles” option, the “a‐NOE” distance inequality constraints and the variable storage first derivative minimization method. The program possibilities are illustrated by examples of molecular docking, energy barrier estimation, modeling of infinite structures, and DNA bending simulations.

A Strange Ultrametric Geometry

The main purpose of this paper is to answer a question raised by Dence in [2]. In that article, he defines a certain kind of metric, called an ultrametric, on the rationals Q and questions whether it can be extended to Q x Q, while still maintaining its special properties. We do just that in this article, creating thereby an unusual geometry that is very close to a model of Euclidean geometry in the sense that most of the Euclidean axioms are satisfied. However, it is quite different from the ordinary model of Euclidean geometry in many ways. Thus it shows that when attempting proofs in axiomatic geometries, one must not rely too much on 'common sense', but rather on straightforward deductions form the axioms and previous theorems. To construct our model, we need to introduce the concept of an ultrametric distance. Recall that a metric on S is a function d: SxS->[O,oo) such that for x, y, zES, (1) d(x,y)=O iff x=y; (2) d(x, y) = d(y, x); and (3) d(x, z) _ d(x, y) + d(y, z). Hence, on the real line R, ordinary distance (i.e., d(a, b) = la b 1) is a metric. To get an ultrametric distance, we replace (3) by the condition (3') d(x, z) _ max(d(x, y), d(y, z)). The reader will easily notice that (3') is a strictly stronger condition than (3) and that the ordinary distance function on R is not an ultrametric. We now describe a particular ultrametric distance function on the set Q of rational numbers. Given x E Q, X7 0, write x = 2k(a/b) where a and b are odd integers. Then the exponent k is uniquely determined. We then define Ix 12 = (1/2)k and d(x, y) = x y 12. If x = 0, then IX = 0. For example, if x = 5/12 = 2-2(5/3), then [x 12 = (1/2)-2 = 4. The verification that this function satisfies the ultrametric inequality can be found in [1]. (There is nothing special about the number 2 in this definition. Any prime number p will work: if x = pk(alb), where a and b are relatively prime to p, then we can define Ix |P = (1/p)k. And even this is a specific case in the general theory of ultrametric spaces. A nice discussion of some of these general abstract ideas can be found in [2].) Dence raised the question as to whether or not we can non-trivially extend this ultrametric to two dimensions. Let us first examine a few possible ways of extending, all of which fail. If A = (a,, a2), B = (b,, b2) E Q X Q, define D1(A, B) = d(al, b1) + d(a2, b2) and D2(A, B) = d(al, b1) (where d is the ultrametric distance on Q described above). Then both of the above functions fail to satisfy the ultrametric distance inequality; in fact D2 is not even metric. A more familiar way to extend a metric is exhibited by D3(P1, P2) = (d(pi, p2)2 + d(ql, q2)2)12. The reader will recognize that this is the same procedure as is used in extending the ordinary distance function on R to the Cartesian plane. Unfortunately, though D3 is a metric, it also does not satisfy the ultrametric inequality. To see this, let A = (1/2,1/2), B = (0, 1), C = (1/3, 2/3); then

The detection and estimation of character weighting in classifications

Relative weighting of characters used in taxonomic decisions is detected by comparison with taxonomic models in which characters are given equal weights. Classifications are analysed for implied distance inequalities between triplets of taxa, and the minimal weights applied to the distances between taxa necessary to satisfy these constraints are estimated. Weights acting as multipliers on interacting characters are compared with weight estimations: geometric parameters which depend upon the relative locations of the taxa in taxonomic space.

Social Distance and Social Inequality in a Ghetto Kindergarten Classroom

Social Distance and Social Inequality in a Ghetto Kindergarten Classroom