Improved upper and lower bounds for the spectral radius of digraphs

Let G be a digraph with n vertices and m arcs without loops and multiarcs. The spectral radius ρ(G) of G is the largest eigenvalue of its adjacency matrix. In this paper, sharp upper and lower bounds on ρ(G) are given. We show that some known bounds can be obtained from our bounds.     

On consistency in probabilistic logic for logical formulas represented by B-hypertrees

One of the problems (mainly unsolved) in probabilistic logic is to consistently assign probabilities to logical formulas. In this paper we consider Horn formulas represented by B-hypertrees. We give a set of necessary conditions that any valid assignment of probabilities to the logical formulas should fulfill. If a certain condition is imposed on the B-hypertree, the necessary conditions are also sufficient, thus describing exactly which rules the assigned probabilities should obey to be consistent.     

Some upper and lower bounds on PSD-rank

Positive semidefinite rank (PSD-rank) is a relatively new complexity measure on matrices, with applications to combinatorial optimization and communication complexity. We first study several basic properties of PSD-rank, and then develop new techniques for showing lower bounds on the PSD-rank. All of these bounds are based on viewing a positive semidefinite factorization of a matrix M as a quantum communication protocol. These lower bounds depend on the entries of the matrix and not only on its support (the zero/nonzero pattern), overcoming a limitation of some previous techniques. We compare these new lower bounds with known bounds, and give examples where the new ones are better. As an application we determine the PSD-rank of (approximations of) some common matrices.     

On lower bounds on the size of sums-of-squares formulas

For sums-of-squares formulas of the form

     

On lower and upper bounds of matrices

Using an approach of Bergh, we give an alternate proof of Bennett's result on lower bounds for non-negative matrices acting on non-increasing non-negative sequences in lp when p?1 and its dual version, the upper bounds when 0<p?1. We also determine such bounds explicitly for some families of matrices.     

Computing upper and lower bounds in interval decision trees

This article presents algorithms for computing optima in decision trees with imprecise probabilities and utilities. In tree models involving uncertainty expressed as intervals and/or relations, it is necessary for the evaluation to compute the upper and lower bounds of the expected values. Already in its simplest form, computing a maximum of expectancies leads to quadratic programming (QP) problems. Unfortunately, standard optimization methods based on QP (and BLP – bilinear programming) are too slow for the evaluation of decision trees in computer tools with interactive response times. Needless to say, the problems with computational complexity are even more emphasized in multi-linear programming (MLP) problems arising from multi-level decision trees. Since standard techniques are not particularly useful for these purposes, other, non-standard algorithms must be used. The algorithms presented here enable user interaction in decision tools and are equally applicable to all multi-linear programming problems sharing the same structure as a decision tree.     

Some upper and lower bounds on the coupon collector problem

The classical coupon collector problem is closely related to probabilistic-packet-marking (PPM) schemes for IP traceback problem in the Internet. In this paper, we study the classical coupon collector problem, and derive some upper and lower bounds of the complementary cumulative distribution function (ccdf) of the number of objects (coupons) that one has to check in order to detect a set of different objects. The derived bounds require much less computation than the exact formula. We numerically find that the proposed bounds are very close to the actual ccdf when detecting probabilities are set to the values common to the PPM schemes.     

Systems of quasi-equilibrium problems with lower and upper bounds

In this work, we consider systems of quasi-equilibrium problems with lower and upper bounds and establish the existence of their solutions by using some known maximal element theorems for a family of multivalued maps. Our problems are more general than the one posed in [G. Isac, V.M. Sehgal, S.P. Singh, An alternative version of a variational inequality, Indian J. Math. 41 (1999) 25–31]. As a particular case, we also get the answer to the problem raised in [G. Isac, V.M. Sehgal, S.P. Singh, An alternative version of a variational inequality, Indian J. Math. 41 (1999) 25–31].     

Computing closely matching upper and lower bounds on textile nesting problems

We consider an industrial cutting problem in textile manufacturing and report on algorithms for computing cutting images and lower bounds on waste for this problem. For the upper bounds we use greedy strategies based on hodographs and global optimization based on simulated annealing. For the lower bounds we use branch-and-bound methods for computing optimal solutions of placement subproblems that determine the performance of the overall subproblem. The upper bounds are computed in less than an hour on a common-day workstation and are competitive in quality with results obtained by human nesters. The lower bounds take a few hours to compute and are within 0.4% of the upper bound for certain types of clothing (e.g., for pants).     

Strict upper and lower bounds of quantities in linear second-order systems

This paper aims to present a model verification technique for general numerical computations of linear second-order systems. Strict upper and lower bounds on quantities of interest are eventually obtained. The model verification technique consists of two major steps. The first is the representation of the error in quantities of interest through an adjoint correction; the second is the application of the strict upper bound derived for the time-discretization error. Moreover, the bounds on quantities of interest are further improved through an optimization procedure. Academic examples are studied to verify the proposed bounds and to explore the potential application of these bounds to adaptive time-stepping.     

A lower and upper bounds version of a variational inequality

Introducing the concept of extremal subset to solve one open problem raised in [1]: find the conditions for a lower and upper bounds version of a variational inequality. A few applications are given.     

The kth upper and lower bases of primitive nonpowerful minimally strong signed digraphs

In this article, we study the kth upper and lower bases of primitive nonpowerful minimally strong signed digraphs. A bound on the kth upper bases for primitive nonpowerful minimally strong signed digraphs is obtained, and the equality case of the bound is characterized. For the kth lower bases, we obtain some bounds. For some cases, the bounds are best possible and the extremal signed digraphs are characterized. We also show that there exist ‘gaps’ in both the kth upper base set and the kth lower base set of primitive nonpowerful minimally strong signed digraphs.     

Separating sets by lower and upper semicontinuous functions with a closed graph

In this paper we characterize the pairs (A^?, A⁺) of disjoint subsets of perfectly normal topological space which can be separated by a lower and an upper semicontinuous function with a closed graph.     

Pointwise lower and upper bounds for eigenfunctions of ordinary differential operators

Summary For a differential operatorL as defined in (2.1) we consider the eigenvalue problemL u=u and describe a method to obtain a pointwise bound |u–v| wherev denotes the Rayleigh-Ritz approximation to an exact eigenfunctionu. The upper bound is continuous, but only piecewise differentiable and calculated by solving certain (inverse-positive!) auxiliary problems.Our method uses well-known estimations for (t _i) at a finite number of pointst _i[a, b] to calculate an upper, bound in the whole interval [a, b].The author would like to, thank the Battelle Institute Geneve for their assistance     

Improved upper and lower bounds on the feedback vertex numbers of grids and butterflies

We improve upon the best known upper and lower bounds on the sizes of minimal feedback vertex sets in butterflies. Also, we construct new feedback vertex sets in grids so that for a large number of pairs (n,m), the size of our feedback vertex set in the grid M_n,m matches the best known lower bound, and for all other pairs it differs from this lower bound by at most 2.     

On the estimates of the upper and lower bounds of Ramanujan primes

For \(n\ge 1\), the nth Ramanujan prime is defined as the least positive integer \(R_{n}\) such that for all \(x\ge R_{n}\), the interval \((\frac{x}{2}, x]\) has at least n primes. Let \(p_{i}\) be the ith prime and \(R_{n}=p_{s}\). Sondow, Laishram, and other scholars gave a series of upper bounds of s. In this paper we establish several results giving estimates of upper and lower bounds of Ramanujan primes. Using these estimates, we discuss a conjecture on Ramanujan primes of Sondow–Nicholson–Noe and prove that if \(n>10^{300}\), then \(\pi (R_{mn})\le m\pi (R_{n})\) for \(m\ge 1\).