Class of tight bounds on the Q‐function with closed‐form upper bound on relative error

In this paper, we propose a novel class of parametric bounds on the Q‐function, which are lower bounds for 1 ≤ a < 3 and x > x_t = (a (a‐1) / (3‐a))^1/2, and upper bound for a = 3. We prove that the lower and upper bounds on the Q‐function can have the same analytical form that is asymptotically equal, which is a unique feature of our class of tight bounds. For the novel class of bounds and for each particular bound from this class, we derive the beneficial closed‐form expression for the upper bound on the relative error. By comparing the bound tightness for moderate and large argument values not only numerically, but also analytically, we demonstrate that our bounds are tighter compared with the previously reported bounds of similar analytical form complexity.     

Estimates of the first Neumann eigenvalue and the log‐Sobolev constant on non‐convex manifolds

In this paper a number of explicit lower bounds are presented for the first Neumann eigenvalue on non‐convex manifolds. The main idea to derive these estimates is to make a conformal change of the metric such that the manifold is convex under the new metric, which enables one to apply known results obtained in the convex case. This method also works for more general functional inequalities. In particular, some explicit lower bounds are presented for the log‐Sobolev constant on non‐convex manifolds. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A parameter‐uniform scheme for the parabolic singularly perturbed problem with a delay in time

In this paper, a parameter‐uniform numerical scheme for the solution of singularly perturbed parabolic convection–diffusion problems with a delay in time defined on a rectangular domain is suggested. The presence of the small diffusion parameter ? leads to a parabolic right boundary layer. A collocation method consisting of cubic B ‐spline basis functions on an appropriate piecewise‐uniform mesh is used to discretize the system of ordinary differential equations obtained by using Rothe's method on an equidistant mesh in the temporal direction. The parameter‐uniform convergence of the method is shown by establishing the theoretical error bounds. The numerical results of the test problems validate the theoretical error bounds.     

Some non‐standard problems in viscous flow

Energy bounds are derived for Dirichlet type boundary value problems for the Navier–Stokes and Stokes equations when a combination of the solution values initially and at a later time is prescribed. The bounds are obtained by means of a differential inequality and imply uniqueness and continuous data dependence of the solutions for a range of values of the parameter in the non‐standard auxiliary condition. Copyright © 2004 John Wiley & Sons, Ltd.     

One‐factorizations of complete graphs with vertex‐regular automorphism groups

We consider one‐factorizations of K_2n possessing an automorphism group acting regularly (sharply transitively) on vertices. We present some upper bounds on the number of one‐factors which are fixed by the group; further information is obtained when equality holds in these bounds. The case where the group is dihedral is studied in some detail, with some non‐existence statements in case the number of fixed one‐factors is as large as possible. Constructions both for dihedral groups and for some classes of abelian groups are given. © 2002 John Wiley & Sons, Inc. J Combin Designs 10: 1–16, 2002     

Tight bounds for popping algorithms

We sharpen run‐time analysis for algorithms under the partial rejection sampling framework. Our method yields improved bounds for: the cluster‐popping algorithm for approximating all‐terminal network reliability; the cycle‐popping algorithm for sampling rooted spanning trees; and the sink‐popping algorithm for sampling sink‐free orientations. In all three applications, our bounds are not only tight in order, but also optimal in30 constants.     

Connected Covering Numbers

A connected covering is a design system in which the corresponding block graph is connected. The minimum size of such coverings are called connected coverings numbers. In this paper, we present various formulas and bounds for several parameter settings for these numbers. We also investigate results in connection with Turán systems. Finally, a new general upper bound, improving an earlier result, is given. The latter is used to improve upper bounds on a question concerning oriented matroid due to Las Vergnas.     

Optimal Gerschgorin‐type inclusion intervals of singular values

In this paper, some optimal inclusion intervals of matrix singular values are discussed in the set Ω_A of matrices equimodular with matrix A. These intervals can be obtained by extensions of the Gerschgorin‐type theorem for singular values, based only on the use of positive scale vectors and their intersections. Theoretic analysis and numerical examples show that upper bounds of these intervals are optimal in some cases and lower bounds may be non‐trivial (i.e. positive) when PA is a H‐matrix, where P is a permutation matrix, which improves the conjecture in Reference (Linear Algebra Appl 1984; 56 :105‐119). Copyright © 2006 John Wiley & Sons, Ltd.     

Normal and tangential continuous elements for least‐squares mixed finite element methods

We consider a finite element discretization of the primal first‐order least‐squares mixed formulation of the second‐order elliptic problem. The unknown variables are displacement and flux, which are approximated by equal‐order elements of the usual continuous element and the normal continuous element, respectively. We show that the error bounds for all variables are optimal. In addition, a field‐based least‐squares finite element method is proposed for the 3D‐magnetostatic problem, where both magnetic field and magnetic flux are taken as two independent variables which are approximated by the tangential continuous and the normal continuous elements, respectively. Coerciveness and optimal error bounds are obtained. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004.     

Finite element superconvergence approximation for one‐dimensional singularly perturbed problems

Superconvergence approximations of singularly perturbed two‐point boundary value problems of reaction‐diffusion type and convection‐diffusion type are studied. By applying the standard finite element method of any fixed order p on a modified Shishkin mesh, superconvergence error bounds of (N^?1 ln (N + 1))^p+1 in a discrete energy norm in approximating problems with the exponential type boundary layers are established. The error bounds are uniformly valid with respect to the singular perturbation parameter. Numerical tests indicate that the error estimates are sharp; in particular, the logarithmic factor is not removable. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 374–395, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10001     

Bounds for Shannon and Zipf‐Mandelbrot entropies

Shannon and Zipf‐Mandelbrot entropies have many applications in many applied sciences, for example, in information theory, biology and economics, etc. In this paper, we consider two refinements of the well‐know Jensen inequality and obtain different bounds for Shannon and Zipf‐Mandelbrot entropies. First of all, we use some convex functions and manipulate the weights and domain of the functions and deduce results for Shannon entropy. We also discuss their particular cases. By using Zipf‐Mandelbrot laws for different parameters in Shannon entropies results, we obtain bounds for Zipf‐Mandelbrot entropy. The idea used in this paper for obtaining the results may stimulate further research in this area, particularly for Zipf‐Mandelbrot entropy.     

Justifying the small‐world phenomenon via random recursive trees

We present a new technique for proving logarithmic upper bounds for diameters of evolving random graph models, which is based on defining a coupling between random graphs and variants of random recursive trees. The advantage of the technique is three‐fold: it is quite simple and provides short proofs, it is applicable to a broad variety of models including those incorporating preferential attachment, and it provides bounds with small constants. We illustrate this by proving, for the first time, logarithmic upper bounds for the diameters of the following well known models: the forest fire model, the copying model, the PageRank‐based selection model, the Aiello‐Chung‐Lu models, the generalized linear preference model, directed scale‐free graphs, the Cooper‐Frieze model, and random unordered increasing k‐trees. Our results shed light on why the small‐world phenomenon is observed in so many real‐world graphs. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 201–224, 2017     

Minimal blow‐up masses and existence of solutions for an asymmetric sinh‐Poisson equation

For a sinh‐Poisson type problem with asymmetric exponents of interest in hydrodynamic turbulence, we establish the optimal lower bounds for the blow‐up masses. We apply this result to construct solutions of mountain pass type on two‐dimensional tori.     

Correction to: “Exponential bounds on the number of designs with affine parameters”

It is well known that the number of designs with the parameters of a classical design having as blocks the hyperplanes in PG(n, q) or AG(n, q), n?3, grows exponentially. This result was extended recently [5] to designs having the same parameters as a projective geometry design whose blocks are the d‐subspaces of PG(n, q), for any 2?d?n ? 1. In this paper, exponential lower bounds are proved on the number of non‐isomorphic designs having the same parameters as an affine geometry design whose blocks are the d‐subspaces of AG(n, q), for any 2≤d≤n ? 1. Exponential bounds are also proved for the number of resolvable designs with these parameters. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:156‐166, 2011     

Synchronization of fractional‐order chaotic systems with disturbances via novel fractional‐integer integral sliding mode control and application to neuron models

In this paper, a novel fractional‐integer integral type sliding mode technique for control and generalized function projective synchronization of different fractional‐order chaotic systems with different dimensions in the presence of disturbances is presented. When the upper bounds of the disturbances are known, a sliding mode control rule is proposed to insure the existence of the sliding motion in finite time. Furthermore, an adaptive sliding mode control is designed when the upper bounds of the disturbances are unknown. The stability analysis of sliding mode surface is given using the Lyapunov stability theory. Finally, the results performed for synchronization of three‐dimensional fractional‐order chaotic Hindmarsh‐Rose (HR) neuron model and two‐dimensional fractional‐order chaotic FitzHugh‐Nagumo (FHN) neuron model.     

A posteriori estimates for the Stokes eigenvalue problem

We consider the Stokes eigenvalue problem. For the eigenvalues we derive both upper and lower a‐posteriori error bounds. The estimates are verified by numerical computations. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Web‐spline‐based finite element approximation of some quasi‐newtonian flows: Existence‐uniqueness and error bound

This article deals with the web‐spline‐based finite element approximation of quasi‐Newtonian flows. First, we consider the scalar elliptic p‐Laplace problem. Then, we consider quasi‐Newtonian flows where viscosity obeys power law or Carreau law. We prove well‐posedness at the continuous as well as the discrete level. We give some error bounds for the solution of quasi‐Newtonian flow problem based on the web‐spline method. Finally, we provide the numerical results for the p‐Laplace problem. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq31: 54–77, 2015     

On existence of complete sets for bounded reducibilities

Classical reducibilities have complete sets U that any recursively enumerable set can be reduced to U. This paper investigates existence of complete sets for reducibilities with limited oracle access. Three characteristics of classical complete sets are selected and a natural hierarchy of the bounds on oracle access is built. As the bounds become stricter, complete sets lose certain characteristics and eventually vanish. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Path homomorphisms,graph colorings,and boolean matrices

We investigate bounds on the chromatic number of a graph G derived from the nonexistence of homomorphisms from some path \begin{eqnarray*}\vec{P}\end{eqnarray*} into some orientation \begin{eqnarray*}\vec{G}\end{eqnarray*} of G. The condition is often efficiently verifiable using boolean matrix multiplications. However, the bound associated to a path \begin{eqnarray*}\vec{P}\end{eqnarray*} depends on the relation between the “algebraic length” and “derived algebraic length” of \begin{eqnarray*}\vec{P}\end{eqnarray*}. This suggests that paths yielding efficient bounds may be exponentially large with respect to G, and the corresponding heuristic may not be constructive. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 198–209, 2010     

Upper and lower Hausdorff dimension estimates for invariant sets of weak k‐1‐endomorphisms

For a special class of non‐injective maps called the weak k‐1‐endomorphisms on Riemannian manifolds upper and lower bounds for Hausdorff dimensions of invariant sets are given in terms of the singular values of the tangent maps, which generalize Franz's corresponding results. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)