Multiquadratic Extensions, Rigid Fields and Pythagorean Fields

Let F be a field of characteristic other than 2. Let F⁽²⁾ denotethe compositum over F of all quadratic extensions of F, letF⁽³⁾ denote the compositum over F⁽²⁾ of all quadratic extensionsof F⁽²⁾ that are Galois over F, and let F^{3} denote the compositumover F⁽²⁾ of all quadratic extensions of F⁽²⁾. This paper showsthat F⁽³⁾ = F^{3} if and only if F is a rigid field, and thatF⁽³⁾ = K⁽³⁾ for some extension K of F if and only if F is Pythagoreanand . The proofs depend mainly on the behavior of quadratic forms over quadratic extensions,and the corresponding norm maps.     

Waldspurger's Involution and Types

Waldspurger's involution for the genuine irreducible supercuspidalrepresentations of SL₂(F) is parametrized in terms of typesin the case F p-adic with p odd. In particular, it is shownthat the in-volution is given by conjugating by an element ofGL₂(F) and twisting one of the defining parameters of an associatedtype by a quadratic character, the relevant parameter beinga character on the norm one elements of a quadratic extension.     

Quadratic Termination Properties of Minimization Algorithms I. Statement and Discussion of Results

There is a family of gradient algorithms (Broyden, 1970) thatincludes many useful methods for calculating the least valueof a function F(x), and some of these algorithms have been extendedto solve linearly constrained problems (Fletcher, 1971). Somenew and fundamental properties of these algorithms are given,in the case that F(x) is a positive definite quadratic function.In particular these properties are relevant to the case whenonly some of the iterations of an algorithm make a completelinear search. They suggest that Goldfarb's (1969) algorithmfor linearly constrained problems has excellent quadratic terminationproperties, and it is proved that these properties are betterthan has been stated in previously published papers. Also anew technique is identified for unconstrained minimization withoutlinear searches.     

Conjugate Directions without Linear Searches

A modified form of the Quasi-Newton family of variable metricalgorithms used in function minimization is proposed that hasquadratic termination without requiring linear searches. Mostmembers of the Quasi-Newton family rely for quadratic terminationon the fact that with accurate linear searches the directionsgenerated, form a conjugate set when the function is quadratic.With some members of the family the convergence of the sequenceof approximate inverse Hessian matrices to the true inverseHessian is also stable. With the proposed modification the samesequence of matrices and the same set of conjugate directionsare generated without accurate linear searches. On a quadratic function the proposal is also related to a suggestionby Hestenes which generates the same set of conjugate directionswithout accurate linear searches. Both methods therefore findthe minimum of an n dimensional quadratic function in at mostn+2 function and gradient calls. On non-quadratic functions the proposal retains the main advantagesclaimed for both the stable Quasi-Newton and Hestenes approaches.It shows promise in that it is competitive with the most efficientunconstrained optimization algorithms currently available.     

General Orthogonal State Variables and Applications to Lyapunov Functions and Feedback Control Systems

A general set of orthogonal state variables is formulated forlinear time-invariant systems both in the time and the frequencydomains. A general control variable u(t) and general initialconditions v(0) are considered. The meaning of orthogonalityin frequency domain is demonstrated. The applications of generalorthogonal state variables to the evaluation of Lyapunov functionsat t =0 and the evaluation of a quadratic performance criterionof feedback control systems are discussed.     

Inner Functions, Bloch Spaces and Symmetric Measures

Schwarz's lemma asserts that analytic mappings from the unitdisc into itself decrease hyperbolic distances. In this paper,inner functions which decrease hyperbolic distances as muchas possible, when one approaches the unit circle, are constructed.Actually, it is shown that a quadratic condition governs thebest decay of the hyperbolic derivative of an inner function.This is related to a result of L. Carleson on the existenceof singular symmetric measures. As a consequence, some resultson composition operators are obtained, bringing out the importanceof the Bloch spaces in this connection. Another consequenceis a uniform way of producing singular measures which are simultaneouslysymmetric and Kahane. 1991 Mathematics Subject Classification:primary 30D50; secondary 30D45, 26A30, 47B38.     

Conjugate Gradient Algorithms: Quadratic Termination without Linear Searches

This paper studies the behaviour of a family of conjugate gradientoptimization algorithms, of which the best known is probablythat introduced in 1964 by Fletcher & Reeves. This familyhas the property that, on a quadratic function, the directionsgenerated by any member of the family are the same set of conjugatedirections providing that, at each iteration, an exact linearsearch is performed. In this paper a modification is introduced that enables thisset of conjugate directions to be generated without any accurateline searches. This enables the minimum of a quadratic functionto be found in, at most, (n+2) gradient evaluations. As themodification only requires the storage of two additional n-vectors,the storage advantage of conjugate gradient algorithms viz-?-vizvariable metric algorithms is maintained. Finally, a numerical study is reported in which the performanceof this new method is compared to that of various members ofthe unmodified family.     

Optimal boundary control problem for n x n infinite-order parabolic lag system

^** Email: haelsaify{at}yahoo.com In this paper, we study the linear quadratic optimal boundarycontrol problem for n x n coupled system of infinite-order parabolicpartial differential equation, in which time-varying lags appearin the state equation and in the Neumann boundary conditionsimultaneously. By Lions scheme, necessary and sufficient conditionof optimality for the Neumann problem with quadratic functionaland constraint control is derived. Finally, several mathematicalexamples for derived optimality conditions are presented.     

Formulas for Liapunov Functions for Systems of Linear Difference Equations

Explicit quadratic Liapunov functions that provide necessaryand sufficient conditions for the asymptotic stability of thesystem of linear difference equations x (t + 1) = Ax(t) areconstructed by transforming the original systems to y (t + 1)= Gy(t), where G is a companion matrix associated with the characteristicpolynomial of A. A necessary and sufficient condition for allroots of the characteristic polynomial to lie in the unit circle|z| < 1 on the complex plane is also derived. 2000 MathematicalsSubject Classification 39A11, 93D05.     

A sphere in a second degree polynomial creeping flow parallel to a wall

Comprehensive results are provided for the creeping flow arounda spherical particle in a viscous fluid close to a plane wall,when the external velocity is parallel to the wall and variesas a second degree polynomial in the coordinates. By linearityof Stokes equations, the solution is a sum of flows for typicalunperturbed flows: a pure shear flow, a ‘modulated shearflow’, for which the rate of shear varies linearly inthe direction normal to the wall, and a quadratic flow. Solutionsconsidered here use the bipolar coordinates technique. Theycomplement the accurate results of Chaoui and Feuillebois (2003)for the pure shear flow. The solution of Goren and O'Neill (1971)for the quadratic flow is reconsidered and a new analyticalsolution is derived for the ambient modulated shear flow. Theperturbed flow fields for these two cases are presented in detailand discussed. Results for the force and torque friction factorsare provided with a 5 x 10^–17 accuracy as a reference.For the quadratic flow, there is a force and a torque on a fixedsphere. A minimum value of the torque is found for a gap ofabout 0·18a, where a is the sphere radius. This minimumis interpreted in term of the corresponding flow structure.For the modulated shear flow, there is only a torque. The freemotion of a sphere in an ambient quadratic flow is also determined.     

On the Multiplicities of Graph Eigenvalues

Star complements and associated quadratic functions are usedto obtain a sharp upper bound for the order of a graph withan eigenspace of prescribed codimension. It is shown that forregular graphs the bound can be reduced by 1, and that thisreduced bound is attained by a regular graph G if and only ifG is an extremal strongly regular graph. 2000 Mathematics SubjectClassification 05C50.     

The numerical solution of a non-linear boundary integral equation on smooth surfaces

We study a boundary integral equation method for solving Laplace'sequation u=0 with non-linear boundary conditions. This non-linearboundary value problem is reformulated as a non-linear boundaryintegral equation, with u on the boundary as the solution beingsought. The integral equation is solved numerically by usingthe collocation method, with piecewise quadratic functions usedas approximations to u. Convergence results are given for thecases where (1) the original surface is used, and (2) the surfaceis approximated by piecewise quadratic interpolation. In addition,we define and analyze a two-grid iteration method for solvingthe non-linear system that arises from the discretization ofthe boundary integral equation. Numerical examples are given;and the paper concludes with a short discussion of the relativecost of different parts of the method. This work was supported in part by NSF grant DMS-9003287.     

Optimal control of n x n parabolic system involving time lag

In this paper, we study the optimal control problem for an nx n coupled system of second-order parabolic partial differentialequations with infinitely many variables and constant time lag.Making use of the Lions scheme (Lions, 1971, Optimal Controlof Systems Governed by Partial Differential Equations), necessaryand sufficient condition of optimality for the Neumann problemwith quadratic performance functional and constraint controlis derived. Finally, several mathematical examples for derivedoptimality conditions are presented.     

Minimization of a Quadratic Function of Many Variables Subject only to Lower and Upper Bounds

Methods for solving this problem are considered with particularreference to achieving maximum efficiency. A streamlined versionof Fletcher's (1971) method for quadratic programming is consideredand also a new approach based on the use of partial LDL^T factorizations.Results on a wide variety of test problems indicate that theLDL^T method is superior in both efficiency and error control.This method can often be expected to solve the problem in atime comparable to that required for a Choleski factorization,and always in a small multiple of this time.     

A Lower Bound for the Cost Functional for Control Problems in Hilbert Space

Linear quadratic problems for evolution equations in Hilbertspace are considered. Given any control u₀, a lower bound forthe ratio of the cost of performance of any other control tothe cost of u₀ is obtained. The significance of this lower boundis discussed and the results applied to some simple examplesof both ordinary and partial differential equations.     

The Minimum Sum of Squares Change to Univariate Data that gives Convexity

Let n measurements of a real valued function of one variablebe given. If the function is convex but the data have lost convexitydue to the errors of the measuring process, then the least sumof squares change to the data that provides nonnegative seconddivided differences may be required. An algorithm is proposedfor this highly structured quadratic programming calculation.First a procedure that requires only O(n) computer operationsgenerates a starting point for the main calculation, and thena version of the iterative method of Goldfarb & Idnani (1983)is applied. It is proved that the algorithm converges, the analysisbeing a special case of the theory of Goldfarb & Idnani.The algorithm is efficient because the matrices that occur arebanded due to representing the required fit as a linear combinationof B-splines. Some numerical results illustrate the method.They suggest that the algorithm can be used when n is very large,because the O(n) starting procedure identifies most of the convexityconstraints that are active at the solution.     

On the Convergence of Some Constrained Minimization Algorithms Based on Recursive Quadratic Programming

This paper considers the convergence of the method of recursiveequality quadratic programming (REQP) for constrained minimization.A theorem of Wolfe (1969) gives conditions on the search directionsand step lengths used by a minimization algorithm which ensurethat it will locate an unconstrained stationary point of a function.It is shown here that, under suitable circumstances, the iterationsof REQP satisfy these conditions both with respect to the conventionalpenalty function P(x, r) and also with respect to the augmentedpenalty function proposed by Fletcher (1969) which has a minimumat the solution to the constrained problem. The behaviour ofREQP in the neighbourhood of the solution is also considered,and it is shown that the algorithm is capable of superlinearconvergence.     

The density of prime divisors in the arithmetic dynamics of quadratic polynomials

Let f [x], and consider the recurrence given by a_n = f(a_{n –1}), with a₀ . Denote by P(f, a₀) the set of prime divisorsof this recurrence, that is, the set of primes dividing at leastone non-zero term, and denote the natural density of this setby D(P(f, a₀)). The problem of determining D(P(f, a₀)) whenf is linear has attracted significant study, although it remainsunresolved in full generality. In this paper, we consider thecase of f quadratic, where previously D(P(f, a₀)) was knownonly in a few cases. We show that D(P(f, a₀)) = 0 regardlessof a₀ for four infinite families of f, including f = x² + k,k \{–1}. The proof relies on tools from group theoryand probability theory to formulate a sufficient condition forD(P(f, a₀)) = 0 in terms of arithmetic properties of the forwardorbit of the critical point of f. This provides an analogy toresults in real and complex dynamics, where analytic propertiesof the forward orbit of the critical point have been shown todetermine many global dynamical properties of a quadratic polynomial.The article also includes apparently new work on the irreducibilityof iterates of quadratic polynomials.     

On the j-Invariants of the Quadratic Q-Curves

The j-invariants of the quadratic Q-curves without complex multiplicationare studied. Some properties of the norms of these invariantsare shown and a relationship between the field Q(j) and thedegree of an isogeny of the Q-curve to its Galois conjugateis found. In the case when the degree of the isogeny is a primep, some properties of the primes of potentially multiplicativereduction for the Q-curve and of the reduction of j modulo aprime P in Q(j) over p when the Q-curve has potentially goodreduction at P are found.     

On weakly eutactic forms

We make precise some properties of the Hermite function in relationwith the Morse theory introduced by Avner Ash in his papers‘On eutactic forms’,Canad. J. Math. 29 (1977) 1040–1054and ‘On the existence of eutactic forms’,Bull. LondonMath. Soc. 12 (1980) 192–196, and with the cellular decompositionof the space of positive definite quadratic forms. We also establisha link between Ash's and Bavard's mass formulae.