The affine Cauchy problem

The aim of this paper is to solve the Cauchy problem for locally strongly convex surfaces which are extremal for the equiaffine area functional. These surfaces are called affine maximal surfaces and here, we give a new complex representation which let us describe the solution to the corresponding Cauchy problem. As applications, we obtain a generalized symmetry principle, characterize when a curve in R³ can be a geodesic or pre-geodesic of a such surface and study the helicoidal affine maximal surfaces. Finally, we investigate the existence and uniqueness of affine maximal surfaces with a given analytic curve in its singular set.     

A solution of the affine quadratic inverse eigenvalue problem

The quadratic inverse eigenvalue problem (QIEP) is to find the three matrices M,C, and K, given a set of numbers, closed under complex conjugations, such that these numbers become the eigenvalues of the quadratic pencil P(λ)=λ²M+λC+K. The affine inverse quadratic eigenvalue problem (AQIEP) is the QIEP with an additional constraint that the coefficient matrices belong to an affine family, that is, these matrices are linear combinations of substructured matrices. An affine family of matrices very often arise in vibration engineering modeling and analysis. Research on QIEP and AQIEP are still at developing stage. In this paper, we propose three methods and the associated mathematical theories for solving AQIEP: A Newton method, an alternating projections method, and a hybrid method combining the two. Validity of these methods are illustrated with results on numerical experiments on a spring-mass problem and comparisons are made with these three methods amongst themselves and with another Newton method developed by Elhay and Ram (2002) [12]. The results of our experiments show that the hybrid method takes much smaller number of iterations and converges faster than any of these methods.     

On the ADI method for Sylvester equations

This paper is concerned with the numerical solution of large scale Sylvester equations AX−XB=C, Lyapunov equations as a special case in particular included, with C having very small rank. For stable Lyapunov equations, Penzl (2000) [22] and Li and White (2002) [20] demonstrated that the so-called Cholesky factor ADI method with decent shift parameters can be very effective. In this paper we present a generalization of the Cholesky factor ADI method for Sylvester equations. An easily implementable extension of Penz’s shift strategy for the Lyapunov equation is presented for the current case. It is demonstrated that Galerkin projection via ADI subspaces often produces much more accurate solutions than ADI solutions.     

Finite termination of a smoothing-type algorithm for the monotone affine variational inequality problem

In this paper, we consider the monotone affine variational inequality problem (AVIP for short). Based on a smooth reformulation of the AVIP, we propose a Newton-type method to solve the monotone AVIP, where a testing procedure is embedded into our algorithm. Under mild assumptions, we show that the proposed algorithm may find a maximally complementary solution to the monotone AVIP in a finite number of iterations. Preliminary numerical results are reported.     

The affine separation problem revisited

The aim of this note is to characterize in terms of inequalities those pairs of real functions (acting on a convex subset of a vector space) that possess an affine separator. The main result is originally due to Behrends and Nikodem. Their method is based on the Hahn–Banach Theorem and a variant of the Helly Theorem. In our approach, a direct and independent proof is presented via the Radon and the Helly Theorems.     

On the explicit solutions of forms of the Sylvester and the Yakubovich matrix equations

In this paper, we consider the explicit solutions of two matrix equations, namely, the Yakubovich matrix equation V−AVF=BW and Sylvester matrix equations AV−EVF=BW,AV+BW=EVF and AV−VF=BW. For this purpose, we make use of Kronecker map and Sylvester sum as well as the concept of coefficients of characteristic polynomial of the matrix A. Some lemmas and theorems are stated and proved where explicit and parametric solutions are obtained. The proposed methods are illustrated by numerical examples. The results obtained show that the methods are very neat and efficient.     

On affine designs and Hadamard designs with line spreads

Rahilly [On the line structure of designs, Discrete Math. 92 (1991) 291-303] described a construction that relates any Hadamard design H on 4^m-1 points with a line spread to an affine design having the same parameters as the classical design of points and hyperplanes in AG(m,4). Here it is proved that the affine design is the classical design of points and hyperplanes in AG(m,4) if, and only if, H is the classical design of points and hyperplanes in PG(2m-1,2) and the line spread is of a special type. Computational results about line spreads in PG(5,2) are given. One of the affine designs obtained has the same 2-rank as the design of points and planes in AG(3,4), and provides a counter-example to a conjecture of Hamada [On the p-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its applications to error-correcting codes, Hiroshima Math. J. 3 (1973) 153-226].     

Testing for affine equivalence of elliptically symmetric distributions

Let X and Y be d-dimensional random vectors having elliptically symmetric distributions. Call X and Y affinely equivalent if Y has the same distribution as AX+b for some nonsingular d×d-matrix A and some . This paper studies a class of affine invariant tests for affine equivalence under certain moment restrictions. The test statistics are measures of discrepancy between the empirical distributions of the norm of suitably standardized data.     

On the elementary divisors of the Sylvester and Lyapunov maps

The paper addresses the problem of computing the elementary divisors of the tensor product of linear transformations using the analysis of the tensor products of polynomial models, as developed in Fuhrmann and Helmke [5]. We use this to study the elementary divisors of the Lyapunov and complementary Lyapunov maps.     

On affine scaling and semi-infinite programming

We consider an extension of the affine scaling algorithm for linear programming problems with free variables to problems having infinitely many constraints, and explore the relationship between this algorithm and the finite affine scaling method applied to a discretization of the problem.This material is based on research supported by Air Force Office of Scientific Research Grant AFOSR 89-0410.     

On affine difference sets and their multipliers

Let D be an affine difference set of order n in an abelian group G relative to a subgroup N. We denote by π(s) the set of primes dividing an integer and set H^∗=H?{ω}, where H=G/N and ω=∏_σ∈Hσ. In this article, using D we define a map g from H to N satisfying for iff {τ,τ⁻¹}={ρ,ρ⁻¹} and show that for any σ∈H^∗ and any integer m>0 with π(m)⊂π(n). This result is a generalization of J.C. Galati’s theorem on even order n [J.C. Galati, A group extensions approach to affine relative difference sets of even order, Discrete Mathematics 306 (2006) 42-51] and gives a new proof of a result of Arasu-Pott on the order of a multiplier modulo exp(H) ([K.T. Arasu, A. Pott, On quasi-regular collineation groups of projective planes, Designs Codes and Cryptography 1 (1991) 83-92] Section 5).     

On the affine Bernstein problem

A partial solution to the affine Bernstein problem is given. The elliptic paraboloid is characterized as a locally strongly convex, affine complete, affine-maximal surface in A ³, satisfying a certain growth condition, about its affine Gauss-Kronecker curvature.Research partially supported by DGICYT Grant PS87-0115-CO3-02.     

The Newton Bracketing method for the minimization of convex functions subject to affine constraints

The Newton Bracketing method [Y. Levin, A. Ben-Israel, The Newton Bracketing method for convex minimization, Comput. Optimiz. Appl. 21 (2002) 213-229] for the minimization of convex functions f:Rⁿ→R is extended to affinely constrained convex minimization problems. The results are illustrated for affinely constrained Fermat-Weber location problems.     

A non-linear structure preserving matrix method for the low rank approximation of the Sylvester resultant matrix

A non-linear structure preserving matrix method for the computation of a structured low rank approximation of the Sylvester resultant matrix S(f,g) of two inexact polynomials f=f(y) and g=g(y) is considered in this paper. It is shown that considerably improved results are obtained when f(y) and g(y) are processed prior to the computation of , and that these preprocessing operations introduce two parameters. These parameters can either be held constant during the computation of , which leads to a linear structure preserving matrix method, or they can be incremented during the computation of , which leads to a non-linear structure preserving matrix method. It is shown that the non-linear method yields a better structured low rank approximation of S(f,g) and that the assignment of f(y) and g(y) is important because may be a good structured low rank approximation of S(f,g), but may be a poor structured low rank approximation of S(g,f) because its numerical rank is not defined. Examples that illustrate the differences between the linear and non-linear structure preserving matrix methods, and the importance of the assignment of f(y) and g(y), are shown.     

Duality theorem for a generalized Fermat-Weber problem

The classical Fermat-Weber problem is to minimize the sum of the distances from a point in a plane tok given points in the plane. This problem was generalized by Witzgall ton-dimensional space and to allow for a general norm, not necessarily symmetric; he found a dual for this problem. The authors generalize this result further by proving a duality theorem which includes as special cases a great variety of choices of norms in the terms of the Fermat-Weber sum. The theorem is proved by applying a general duality theorem of Rockafellar. As applications, a dual is found for the multi-facility location problem and a nonlinear dual is obtained for a linear programming problem with a priori bounds for the variables. When the norms concerned are continuously differentiable, formulas are obtained for retrieving the solution for each primal problem from the solution of its dual.     

Principal bundles on the affine line

We prove that any principal bundle on the affine line over a perfect field with a reductive group as structure group comes from the base field by base change.     

An improved non-linear method for the computation of a structured low rank approximation of the Sylvester resultant matrix

This paper reports on improvements to recent work on the computation of a structured low rank approximation of the Sylvester resultant matrix S(f,g)$S(f,g)$ of two inexact polynomials f=f(y)$f=f\left(y\right)$ and g=g(y)$g=g\left(y\right)$. Specifically, it has been shown in previous work that these polynomials must be processed before a structured low rank approximation of S(f,g)$S(f,g)$ is computed. The existing algorithm may still, however, yield a structured low rank approximation of S(f,g)$S(f,g)$, but not a structured low rank approximation of S(g,f)$S(g,f)$, which is unsatisfactory. Moreover, a structured low rank approximation of S(f,g)$S(f,g)$ must be equal to, apart from permutations of its columns, a structured low rank approximation of S(g,f)$S(g,f)$, but the existing algorithm does not guarantee the satisfaction of this condition. This paper addresses these issues by modifying the existing algorithm, such that these deficiencies are overcome. Examples that illustrate these improvements are shown.     

Probabilistic properties of the dual structure of the multidimensional knapsack problem and fast statistically efficient algorithms

Properties of several dual characteristics of the multidimensional knapsack problem (such as the probability of existence of-optimal and optimal-feasible Lagrange function generalized saddle points, magnitude of relative duality gap, etc.) are investigated for different probabilistic models. Sufficient conditions of good asymptotic behavior of the dual characteristics are given. A fast statistically efficient approximate algorithm with linear running time complexity for problems with random coefficients is presented.This paper was written when the author was affiliated with Chelyabinsk State Technical University and the Moscow Physical and Technical Institute, Russia.     

On a Nirenberg-type problem involving the square root of the Laplacian

Through an Euler–Hopf-type formula, we establish existence result to a Nirenberg-type problem involving the square root of the Laplacian in sphere S²$S^2$.