On the perturbation of the Q‐factor of the QR factorization

This paper gives normwise and componentwise perturbation analyses for the Q‐factor of the QR factorization of the matrix A with full column rank when A suffers from an additive perturbation. Rigorous perturbation bounds are derived on the projections of the perturbation of the Q‐factor in the range of A and its orthogonal complement. These bounds overcome a serious shortcoming of the first‐order perturbation bounds in the literature and can be used safely. From these bounds, identical or equivalent first‐order perturbation bounds in the literature can easily be derived. When A is square and nonsingular, tighter and simpler rigorous perturbation bounds on the perturbation of the Q‐factor are presented. Copyright © 2011 John Wiley & Sons, Ltd.     

Upper and Lower Bounds on Maximum Nonlinearity of n-input m-output Boolean Function

The paper presents lower and upper bounds on the maximumnonlinearity for an n-input m-output Booleanfunction. We show a systematic construction method for a highlynonlinear Boolean function based on binary linear codes whichcontain the first order Reed-Muller code as a subcode. We alsopresent a method to prove the nonexistence of some nonlinearBoolean functions by using nonexistence results on binary linearcodes. Such construction and nonexistence results can be regardedas lower and upper bounds on the maximum nonlinearity. For somen and m, these bounds are tighter than theconventional bounds. The techniques employed here indicate astrong connection between binary linear codes and nonlinear n-input m-output Boolean functions.     

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.     

Linear Arboricity and Linear k-Arboricity of Regular Graphs

 We find upper bounds on the linear k-arboricity of d-regular graphs using a probabilistic argument. For small k these bounds are new. For large k they blend into the known upper bounds on the linear arboricity of regular graphs. Received: December 21, 1998 Final version received: July 26, 1999     

Automorphism groups of compact bordered Klein surfaces with invariant subsets

In this work we get upper bounds for the order of a group of automorphisms of a compact bordered Klein surface S of algebraic genus greater than 1. These bounds depend on the algebraic genus of S and on the cardinals of finite subsets of S which are invariant under the action of the group. We use our results to obtain upper bounds for the order of a group of automorphism whose action on the set of connected components of the boundary of S is not transitive. The bounds obtained this way depend only on the algebraic genus of S. The author is partially supported by the European Network RAAG HPRN-CT-2001-00271 and the Spanish GAAR DGICYT BFM2002-04797.     

On the index of the Weierstrass semigroup of a pair of points on a curve

We obtain the exact formulas for the cardinality of the complement of the Weierstrass semigroup of a pair (p, q) of points on a curveC. Using these formulas we obtain lower bounds and upper bounds on the cardinalities of such sets. Moreover, considering examples, we show that these bounds are sharp.Partially supported by Korea Science and Engineering Foundation and by Global Analysis Research Center.     

Square function and heat flow estimates on domains

The first purpose of this note is to provide a proof of the usual square function estimate on L^p(Ω). It turns out to follow directly from a generic Mikhlin multiplier theorem obtained by Alexopoulos, and we provide a sketch of its proof in the Appendix for the reader’s convenience. We also relate such bounds to a weaker version of the square function estimate which is enough in most instances involving dispersive PDEs and relies on Gaussian bounds on the heat kernel (such bounds are the key to Alexopoulos’result as well). Moreover, we obtain several useful L^p(Ω;H) bounds for (the derivatives of) the heat flow with values in a given Hilbert space H.     

Transience and capacity of minimal submanifolds

We prove explicit lower bounds for the capacity of annular domains of minimal submanifolds P ^m in ambient Riemannian spaces N ⁿ with sectional curvatures bounded from above. We characterize the situations in which the lower bounds for the capacity are actually attained. Furthermore we apply these bounds to prove that Brownian motion defined on a complete minimal submanifold is transient when the ambient space is a negatively curved Hadamard-Cartan manifold. The proof stems directly from the capacity bounds and also covers the case of minimal submanifolds of dimension m > 2 in Euclidean spaces.     

Improving Hadamard's inequality *

We study several bounds for the determinant of an n × n positive definite Hermitian matrix A. These bounds are the best possible given certain data about A. We find the best bounds in the cases that we are given: (i) the diagonal elements of A: (ii) the traces trA,tr A ² and n and (iii)n, tr A tr A ² and the diagonal elements of A. In case (i) we get the well known Hadamard inequality. The other bounds are Hadamard type bounds. The bounds are found using optimization techniques.     

Global Error Bounds for <Emphasis Type="Italic">γ</Emphasis>-paraconvex Multifunctions

In this paper, error bounds for γ-paraconvex multifunctions are considered. A Robinson-Ursescu type Theorem is given in normed spaces. Some results on the existence of global error bounds are presented. Perturbation error bounds are also studied.