Trace and spectrum preserving linear mappings in Jordan-Banach algebras

In the first section we define the trace on the socle of a Jordan-Banach algebra in a purely spectral way and we prove that it satisfies several identities. In particular this trace defines the Faulkner bilinear form. In the second section, using analytic tools and the properties of the trace, we prove that a spectrum preserving linear mapping fromJ ontoJ ¹, whereJ andJ ¹ are semisimple Jordan-Banach algebras, is not far from being a Jordan isomorphism. It is in particular a Jordan isomorphism ifJ ¹ is primitive with non-zero socle.     

Infinite Simple (2, 3, n)-Groups and Congruence Hulls in the Modular Group

We prove that if n > 66 and (n, 30) = 1, then there existuncountably many infinite simple (2, 3, n)- groups, that is,groups generated by a pair of elements x, y, say, where theorders of x, y and xy are 2, 3 and n, respectively. This extendsprevious results of Schupp and the authors. These results are used to prove the existence of subgroups ofthe modular group with special arithmetic properties. 1991 MathematicsSubject Classification 20F06.     

Structural Stability of Constrained Polynomial Systems

The structural stability of constrained polynomial differentialsystems of the form a(x, y)x'+b(x, y)y'=f(x, y), c(x, y)x'+d(x,y)y'=g(x, y), under small perturbations of the coefficientsof the polynomial functions a, b, c, d, f and g is studied.These systems differ from ordinary differential equations at‘impasse points’ defined by ad–bc=0. Extensionsto this case of results for smooth constrained differentialsystems [7] and for ordinary polynomial differential systems[5] are achieved here. 1991 Mathematics Subject Classification34C35, 34D30.     

Volterra Convolution Operators with Values in Rearrangement Invariant Spaces

The Volterra convolution operator Vf(x) = ^x₀(x–y)f(y)dy,where (·) is a non-negative non-decreasing integrablekernel on [0, 1], is considered. Under certain conditions onthe kernel , the maximal Banach function space on [0, 1] onwhich the Volterra operator is a continuous linear operatorwith values in a given rearrangement invariant function spaceon [0, 1] is identified in terms of interpolation spaces. Thecompactness of the operator on this space is studied.     

The sound field due to a periodic array of forced wave-bearing strips

A compressible fluid in a two-dimensional half-space (y >0) is bounded by a plane surface (y = 0) which is acousticallyhard except for a set of periodically arranged strips S_n givenby nd – a < x < nd + a, y = 0 with n = 0, 1, 2,....The velocity potential Re {(x, y)exp(–it)} satisfies theHelmholtz wave equation in the fluid region y>0, with /y= 0 on the plane y = 0, x S_n. The boundary condition on thepistons S_n is taken to have the form where the prescribed forcing function V(x) is the same on eachstrip, so that V(x + nd) = V(x), and the operators L and M arepolynomial functions of the second derivative ²/x². This boundarycondition includes the possibilities of an elastic plate, amembrane, or an impedance surface for S_n. When the separationdistance d is much greater than the strip width 2a and wavelength2/k, the problem is reduced to that of finding the potential_p due to a single piston S_o set in a rigid baffle, togetherwith a potential _c subject to a similar condition with forcingfunctions exp (ikx) in place of V(x). The problem is generalizedto allow for the possibility of a phased forcing function V(x),such that V(x + nd) = exp (ißnd)V(x), where ßis a given constant.     

Integrals and Areas Expressed as Bilinear Forms of Discrete Defining Data

An ordered set of discrete data-pairs (x_k, y_k) is supposed givenfor k = 1(1)n, derived perhaps as a consequence of experiment,and indicating some form of relationship between x and y. Itis required that these data be interpolated in some systematicway so as to establish this relationship in the form of a rectifiablecurve in the (x, y) plane, and that y is then to be integratedwith respect to x (or vice versa), thereby defining an areain this plane. We shall here consider interpolation schemeswhere by this integral can be expressed as a bilinear form _jKC_jkY_jx_k,the "weighting" coefficients C_jk being numbers independent ofthe data values (though not necessarily of their number, n). The expressions obtained sepcialize to Gregory formulae, andti Simpson's Rule and other forms of Cotes formulae in suitablecontexts, but the information is primarily of use in dealingwith unequally-spaced data values, or in estumating the areawithin a closed curve. The simplicity of the numerical algorithmis clearly unaffected by the geometric complexity of the interpolatedcurve. Values for the weighting coefficients are proposed, and theresults of a numerical experiment are described which teststhe applicability of the formulae described.     

Quasi-Permutation Representations of p-Groups of Class 2

If G is a finite linear group of degree n, that is, a finitegroup of automorphisms of an n-dimensional complex vector space(or, equivalently, a finite group of non-singular matrices oforder n with complex coefficients), we shall say that G is aquasi-permutation group if the trace of every element of G isa non-negative rational integer. The reason for this terminologyis that, if G is a permutation group of degree n, its elements,considered as acting on the elements of a basis of an n-dimensionalcomplex vector space V, induce automorphisms of V forming agroup isomorphic to G. The trace of the automorphism correspondingto an element x of G is equal to the number of letters leftfixed by x, and so is a non-negative integer. Thus, a permutationgroup of degree n has a representation as a quasi-permutationgroup of degree n. See [8].     

On the Comparison of Two Numerical Methods for Conformal Mapping

Let G be a simply connected domain in the t plane (t = x + iy),bounded by the three straight lines x = 0, y = 0, x = 1 anda Jordan arc with cartesian equation y = (x). Also, let g bethe function which maps conformally a rectangle R onto G, sothat the four corners of R are mapped onto those of G. In thispaper we show that the method considered in 1982 by Challis& Burley for determining approximations to g is equivalentto a special case of the well-known method of Garrick for themapping of doubly connected domains. Hence, by using resultsalready available in the literature, we provide some theoreticaljustification for the method of Challis & Burley.     

Primes Represented by Binary Cubic Forms

Let f(x, y) be a binary cubic form with integral rational coefficients,and suppose that the polynomial f(x, y) is irreducible in Q[x,y] and no prime divides all the coefficients of f. We provethat the set f Z⁽²⁾ contains infinitely many primes unless f(a,b) is even for each (a,b) in Z², in which case the set contains infinitely many primes. 2000Mathematical Subject Classification: primary 11N32; secondary11N36, 11R44.     

On the complexity of paths avoiding forbidden pairs

Given a graph G=(V,E), two fixed vertices s,t∈V and a set F of pairs of vertices (called forbidden pairs), the problem of a path avoiding forbidden pairs is to find a path from s to t that contains at most one vertex from each pair in F. The problem is known to be NP-complete in general and a few restricted versions of the problem are known to be in P. We study the complexity of the problem for directed acyclic graphs with respect to the structure of the forbidden pairs.We write x?y if and only if there exists a path from x to y and we assume, without loss of generality, that for every forbidden pair (x,y)∈F we have x?y. The forbidden pairs have a halving structure if no two pairs (u,v),(x,y)∈F satisfy v?x or v=x and they have a hierarchical structure if no two pairs (u,v),(x,y)∈F satisfy u?x?v?y. We show that the PAFP problem is NP-hard even if the forbidden pairs have the halving structure and we provide a surprisingly simple and efficient algorithm for the PAFP problem with the hierarchical structure.     

A Characterization of Finite Soluble Groups by Laws in Two Variables

Define a sequence (s_n) of two-variable words in variables x,y as follows: s₀(x, y) = x, s_n+1(x,y)=[s_n(x, y]^–y, s_n(x,y)for n 0. It is shown that a finite group G is soluble if andonly if s_n is a law of G for all but finitely many values ofn. 2000 Mathematics Subject Classification 20D10, 20D06.     

Commuting holomorphic maps on the spectral unit ball

We prove that if F is a holomorphic map from the open spectralunit ball of a primitive Banach algebra into itself satisfyingF(0) = 0, F^' (0) = I and F(x) x = xF(x) for every x, then Fis the identity map. Using this, we prove that if is a semisimpleBanach algebra and is a primitive Banach algebra, then anyunital spectral isometry from onto which locally preservescommutativity is a Jordan morphism. The same is true when and are both assumed to be von Neumann algebras.     

On Second Order Bifurcations of Limit Cycles

The paper derives a formula for the second variation of thedisplacement function for polynomial perturbations of Hamiltoniansystems with elliptic or hyperelliptic Hamiltonians H(x, y)=y²–U(x)in terms of the coefficients of the perturbation. As an application,the conjecture stated by C. Chicone that a specific cubic systemappearing in a deformation of singularity with two zero eigenvalueshas at most two limit cycles is proved.     

Transmission of sound through a pair of rectangular elastic plates

^** Email: f.leppington{at}ukonline.co.uk Two thin elastic plates, which may have different physical parameters,occupy the parallel regions 0 < x < a, 0 < y < b,z = – d and 0 < x < a, 0 < y < b, z = 0, ina Cartesian coordinate system, where the separation distanced is much less than the dimensions a and b. The plates are simplyconnected to a structure that occupies the rest of the region– d < z < 0. The side walls (x = 0 or a, 0 <y < b, – d < z < 0 and y = 0 or b, 0 < x <a, – d < z < 0) are taken to be acoustically soft;the surfaces z = – d and z = 0, outside the rectangularregions occupied by the plates, are taken to be acousticallyhard. Light compressible fluid (air) is in each of the half-spacesoutside the plates and in the rectangular region between them.A sound field is induced by a high-frequency plane wave. Allowanceis made for a small element of damping in the fluid trappedbetween the plates, as well as damping in the plates. Estimatesare derived for the transmitted power, averaged over a frequencyband and with respect to all incidence directions, in the asymptoticlimits of small fluid loading and high frequency. For the casewhere the two plates have identical physical properties, detailedasymptotic estimates are given for the form of the peaks inthe transmitted power, at a discrete set of operating frequencies,both above and below the plates' coincidence frequency.     

High-Order Embedded Runge-Kutta-Nystrom Formulae

New efficient embedded Runge-Kutta-Nystrom processes of orders8(6) and 12(10) are presented for the numerical solution ofthe special second-order differential equation y'(x) = f[x,y(x)]. Test results indicate their improved efficiency relativeto other RKN formulae in current use.     

Observability of Nonlinear Systems

We consider the observability of systems of the form = Ax +Nx, y = Fx, where A is a linear operator and N and F are nonlinear.We show that if the system is linearized about an equilibriumpoint x_e and the linearized system is continuously initiallyobservable, then the nonlinear system is continuously initiallyobservable in some neighbourhood of x_e. We then look at conditionsunder which solutions of the nonlinear system can be extendedfor all time and consider the problem of stabilizing the systemby feedback controls such that the solutions are eventuallyin the observability neighbourhood of x_e. Finally, we applythese ideas to two systems: a wave equation and a diffusionequation with nonlinear perturbations and nonlinear observations.     

On a Normal Form for Non-Weakly Sequentially Continuous Polynomials on Banach Spaces

Let p be an m-homogeneous polynomial on a complex Banach space,and let (x_n)n be a bounded sequence such that when evaluatedin polynomials of degree less than m, it converges to zero,but p(x_n = 1. It is proved here that there exists a basic sequence(y_k)k equivalent to a subsequence , for which . 2000 MathematicsSubject Classification 46B99.     

Regularity of Rees Algebras

Let B = k[x₁, ..., x_n] be a polynomial ring over a field k,and let A be a quotient ring of B by a homogeneous ideal J.Let m denote the maximal graded ideal of A. Then the Rees algebraR = A[m t] also has a presentation as a quotient ring of thepolynomial ring k[x₁, ..., x_n, y₁, ..., y_n] by a homogeneousideal J^*. For instance, if A = k[x₁, ..., x_n], then Rk[x₁,...,x_n,y₁,...,y_n]/(x_iy_j–x_jy_i|i, j=1,...,n). In this paper we want to compare the homological propertiesof the homogeneous ideals J and J^*.