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.     

A Class of Pencils of Matrices

A linear machine is one in which the time dependent input yis related to the output z by P(D). z = S(D). y where P andS are polynomials in D = d/dt with constant coefficients. Fornumerical computation it is necessary to replace this relationby a set of simultaneous first order differential equationsand this paper shows how to construct such equations by methodswhich extend the results of Gilder (1961). Attention is restrictedto those sets of equations that are of a special form (see (1))which is characterized by the matrix operating on the dependentvariables. This matrix forms a pencil, being linear in D, andthree theorems are given to show how such matrix pencils maybe constructed from the polynomials. The theorems also statethat any matrix pencil with the required properties can be transformedinto the canonical forms given in the theorems by pre- and post-multiplicationby suitable constant non-singular matrices. Thus the variablesof any set of equations having the required properties are linearcombinations of the variables of the equations given by thetheorems. In the paper it is assumed that the degree of P(D)is greater than that of S(D), as otherwise z would be replacedby z₁+Q(D) . y, where Q is the quotient of S(D)/P(D). Also,as the algebriac manipulations are independent of the natureof the polynomials, D is replaced by an indeterminate x andthe coefficients considered to be from an arbitrary field. Fortechnical reasons we rename y and z, y_o and y_n–_m respectively.     

Irregularities of Point Distribution Relative to Convex Polygons III

Suppose that P is a distribution of N points in the unit squareU=[0, 1]². For every x=(x₁, x₂)U, let B(x)=[0, x₁]x[0, x₂] denotethe aligned rectangle containing all points y=(y₁, y₂)U satisfying0y₁x₁ and 0y₂x₂. Denote by Z[P; B(x)] the number of points ofP that lie in B(x), and consider the discrepancy function D[P; B(x)]=Z[P; B(x)]–Nµ(B(x)), where µ denotes the usual area measure.     

Nonoscillation of Elliptic Integrals Related to Cubic Polynomials with Symmetry of Order Three

We study zeros of elliptic integrals I(h)=_HhR(x,y)dxdy, whereH(x,y) is a real cubic polynomial with a symmetry of order three,and R(x,y) is a real polynomial of degree at most n. It turnsout that the vector space A_n formed by such integrals is a Chebishevsystem: the number of zeros of each elliptic integral I(h)A_nis less than the dimension of the vector space A_n. 1991 MathematicsSubject Classification 34C10.     

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.     

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^*.     

Reduction of CalderON Commutators

Let A₁,..., A_n be Lipschitz functions on R such that A'₁,...,A'_nVMO. We show that on any bounded interval, the Calderóncommutator associated with the kernel (A₁(x)–A₁(y)) ...(A_n(x) – A_n(y))/(x–y) ⁿ¹ is a compact perturbationof , where H is the Hilberttransform. 1991 Mathematics Subject Classification 47B38, 47B47,47G10, 45E99.     

SCALED ASYMPTOTICS FOR SOME q-SERIES

This work, investigates the asymptotics for Euler’s q-exponentialE_q(z), Ramanujan’s function A_q(z), Jackson’s q-Besselfunction J_v⁽²⁾ (z; q), the Stieltjes–Wigert orthogonalpolynomials S_n(x; q) and q-Laguerre polynomials L_n⁽⁾ (x; q)as q approaches 1.     

Coherent Products in a Finite Group along a Linear Ordering

Let C = (C, ) be a linear ordering, E a subset of {(x, y):x< y in C} whose transitive closure is the linear orderingC, and let :E G be a map from E to a finite group G = (G, •).We showed with M. Pouzet that, when C is countable, there isF E whose transitive closure is still C, and such that (p) = (x_o, x₁)•(x₁, x₂)•....•(x_{n– 1}, x_n) G depends only upon the extremities x₀, x_n ofp, where p = (x_o, x₁...,x_n) (with 1 n < ) is a finite sequencefor which (x_i, x_{i + 1}) F for all i < n. Here, we show thatthis property does not hold if C is the real line, but is stilltrue if C does not embed an ₁-dense linear ordering, or evena 2-dense linear ordering when Martin's Axiom holds (it followsin particular that it is independent of ZFC for linear orderingsof size ). On the other hand, we prove that this property isalways valid if E = {(x,y):x < y in C}, regardless of anyother condition on C.     

Necessary and Sufficient Conditions for Existence and Uniqueness of Solutions of Second-Order Autonomous Differential Equations

The main result ensures that the scalar problem x' = f(x),x(0) = x₀, x'(0) = x₁, has a nonconstant locally W^{2, 1} solutionif and only if there exists a nontrivial interval J such thatx₀ J, for almost all y Jand Necessary and sufficient conditions for local and global uniquenessand for existence of periodic solutions are also established.     

On Ratio Inequalities for Heat Content

Let U be a domain, convex in x and symmetric about the y-axis,which is contained in a centered and oriented rectangle S. Itis proved that H_t(U⁺)/H_t(U)H_t(S⁺)/H_t(S) where H_t stands forheat content, that is, the remaining heat in the domain at timet if it initially has uniform temperature 1, with Dirichletboundary conditions, where A⁺=A{(x,y):x>0}. It is also shownthat the analog of this inequality holds for some other Schrödingeroperators.     

Simultaneous determination of the source terms in a linear hyperbolic problem from the final overdetermination: weak solution approach

The problem of determining the pair w:={F(x, t);f(t)} of sourceterms in the hyperbolic equation u_tt = (k(x)u_x)_x + F(x, t) andin the Neumann boundary condition k(0)u_x(0, t) = f(t) from themeasured data µ(x):=u(x, T) and/or (x):=u_t(x, t) at thefinal time t = T is formulated. It is proved that both componentsof the Fréchet gradient of the cost functionals J₁(w)= ||u(x, t;w) – µ(x)||₀² and J₂(w) = ||u_t(x, T;w)– (x)||₀² can be found via the solutions of correspondingadjoint hyperbolic problems. Lipschitz continuity of the gradientis derived. Unicity of the solution and ill-conditionednessof the inverse problem are analysed. The obtained results permitone to construct a monotone iteration process, as well as toprove the existence of a quasi-solution.     

A Necessary and Sufficient Condition for a Linear Differential System to be Strongly Monotone

In order to present the results of this note, we begin withsome definitions. Consider a differential system [formula] where IR is an open interval, and f(t, x), (t, x)IxRⁿ, is acontinuous vector function with continuous first derivativesf_r/x_s, r, s=1, 2, ..., n. Let D_xf(t, x), (t, x)IxRⁿ, denote the Jacobi matrix of f(t,x), with respect to the variables x₁, ..., x_n. Let x(t, t₀,x₀), tI(t₀, x₀) denote the maximal solution of the system (1)through the point (t₀, x₀)IxRⁿ. For two vectors x, yRⁿ, we use the notations x>y and x>>yaccording to the following definitions: [formula] An nxn matrix A=(a_rs) is called reducible if n2 and there existsa partition [formula] (p1, q1, p+q=n) such that [formula] The matrix A is called irreducible if n=1, or if n2 and A isnot reducible. The system (1) is called strongly monotone if for any t₀I, x₁,x₂Rⁿ [formula] holds for all t>t₀ as long as both solutions x(t, t₀, x_i),i=1, 2, are defined. The system is called cooperative if forall (t, x)IxRⁿ the off-diagonal elements of the nxn matrix D_xf(t,x) are nonnegative. 1991 Mathematics Subject Classification34A30, 34C99.     

Similarity Solutions of a Non-linear Diffusion Equation

In several physical contexts the equations for the dispersionof a buoyant contaminant can be approximated by the Erdogan-Chatwin(1967) equation {dot}c = {dot}_y{[D_o + ({dot}_yc)²D₂]{dot}_yc}. Here it is shown that in the limit of strong non-linearity (i.e.D_o = 0) there are similarity solutions for a concentration jumpand for a finite discharge. A stability analysis for the latterproblem involves a new family of orthogonal polynomials Y_n(z)where (1 – z⁴)Y – 6z³Y + n(n + 5)z² Y_n = 0 and the degree n is restricted to the values 0, 1, 4, 5, 8,9,.... A numerical solution of the Erdogan-Chatwin equationis given which describes the transition between the non-linearand linear (Gaussian) similarity solutions.     

On Positive Multipeak Solutions of a Nonlinear Elliptic Problem

In this paper we continue our investigation in [5, 7, 8] onmultipeak solutions to the problem –²u+u=Q(x)|u|^q–2u, xR^N, uH¹(R^N) (1.1) where = ^N_i=1²/x²_i is the Laplace operator in R^N, 2 < q < for N = 1, 2, 2 < q < 2N/(N–2) for N3, and Q(x)is a bounded positive continuous function on R^N satisfying thefollowing conditions. (Q₁) Q has a strict local minimum at some point x₀R^N, that is,for some > 0 Q(x)>Q(x₀) for all 0 < |x–x₀| < . (Q₂) There are constants C, > 0 such that |Q(x)–Q(y)|C|x–y| for all |x–x₀| , |y–y₀| . Our aim here is to show that corresponding to each strict localminimum point x₀ of Q(x) in R^N, and for each positive integerk, (1.1) has a positive solution with k-peaks concentratingnear x₀, provided is sufficiently small, that is, a solutionwith k-maximum points converging to x₀, while vanishing as 0 everywhere else in R^N.     

Gauge Fixing for Logarithmic Connections Over Curves and the Riemann-Hilbert Problem

Let be a compact Riemann surface of genus g, X={x₁, ..., x_n} a finite set of points, and ¹(log X) be the sheaf of 1-forms,holomorphic over \X and generated near x_j by dz_j/z_j for a coordinatez_j centred at x_j.     

Asymptotic Expansions of Double and Multiple Integrals Occurring in Diffraction Theory

At present at I.N.S.T.N., Saclay and Faculté des Sciences, Paris, France Asymptotic expansions of double integrals of the type have been derived in terms of thereal parameter k by the method of stationary phase. The resultscan easily be extended to multi-dimensional integrals. In the first part of this paper a rigorous proof of the applicationof the method of stationary phase to double and multiple integralsis established with the aid of neutralizer or unitary functions.It is shown that the principal contributions to U(k) come fromsmall but otherwise arbitrary neighbourhoods of critical pointsof the integral, which may be located in the interior or onthe boundary of the domain of integration. These points areassociated with the phase or amplitude function. An explicitasymptotic series in the parameter k of the principal contributionis exhibited when the amplitude and the phase functions havein the neighbourhood of a critical point (x₁,y₁) a developmentof the form g(x,y) = (x–x₁)^0–1 (y–y₁)^µ0–1g₁(x,y), (x,y) = (x₁,y₁) + a _,0 (x–x₁ [1 + P(x,y) + b_0,(y–y₁[1+Q(x,y)]. The function g₁ is a regular function and P,Q can be developedin power series in the vicinity of the critical point and vanishat this point. The above expansion we shall call normal or canonicaland the critical point a normal or canonical critical pointof the integral. Although the assumption of the normal form expansion of theamplitude and phase functions is too restrictive for the generalcase, nevertheless it is found to be sufficiently broad to includemost of the important and interesting cases which occur in diffraction,scattering and other problems of mathematical physics. In Part II the principal contribution arising from a criticalpoint of normal type has been calculated in the form of a descendingpower series in the parameter k. It is shown, with the use ofmajorant functions, that the contribution due to the remainderpart of the series is of higher order in the parameter thanthat of the last term of the finite part, which proves the asymptoticcharacter of the series in the sense of Poincaré. Theresults derived here are in agreement with that of Part I. However,the new series has a decided advantage over that given in PartI if calculations are desired for even a few terms of the series,since the coefficients entering in the asymptotic expansionof the principal contribution are expressed directly in termsof the original functions g(x,y) and (x,y) and their derivatives,which is not the case in the formulas derived in Part I. In Part III explicit asymptotic expansions of the double integralare derived for several typical critical points associated withthe phase function. These are important in connection with thetheory of diffraction of optical instruments with large aberrationsand scattering problems. On account of their importance, eachcase has been treated in detail. In the appendices we have given an alternative proof of thetheorem announced in Part I and the derivation of the leadingterm due to a boundary stationary point. There will be foundalso a discussion of the more general integral where the parameterk appears implicitly in the phase function and not explicitlyas considered in the text. Integrals of this kind occur in manybranches of physics, especially when dealing with wave propagationin dispersive and absorbing media. Finally, we have concludedon the basis of our results that the Rubinowicz approach todiffraction and the stationary phase application to diffractionintegrals lead to similar mathematical results, although differentphysical interpretations, in diffraction phenomena, the formerleading to Young diffraction phenomena and the latter to Fresneldiffraction phenomena.     

Off-Diagonal Bounds of Non-Gaussian Type for the Dirichlet Heat Kernel

The paper considers the heat kernel K(t, x, y) of the operator– on a proper Euclidean domain , with Dirichlet boundaryconditions. A general pointwise lower bound for K, which isvalid for t larger than a suitable t₀(x,y), is proved (the short-timebehaviour being well understood). The resulting non-Gaussianbounds describe simultaneously both the case of bounded domainsand the case, modelled on the half-space example, of domainswhich satisfy a twisted infinite internal cone condition. Boundsfor the Green's function are given as well.     

On the L-Function of the Curves y2 = x5 + A

Let C_A/Q be the curve y² = x⁵ + A, and let L(s, J_A) denote theL-series of its Jacobian. Under the assumption that the signin the functional equation for L(s, J_A) is +1, the criticalvalue L(1, J_A) is evaluated in terms of the value of a thetaseries for depending on Aat a complex multiplication point coming from Q(₅).