Generating Examples of Shock Propagation by Three Principles

In a medium characterized by a scalar speed C(x), a shock arrivesat the point x, after time T(x), with its magnitude decreasedby A(x). Symmetric C, T, and A in two dimensions can be convertedto cylindrically symmetric results in three dimensions by applyinga dimension-increasing principle: "Let C(x, y), T(x, y), andA(x, y) be even functions of y. They can be extended into threedimensions by using the formulas C(x, y)C(x, r), T(x, y)T(x,r), and A(x,y)A(x,r) [r^–1 cos(x, r)]^?, where r = (x²+2²)^?and is an auxiliary function." When C(x) is a function of asingle variable, the auxiliary function is given by cos(x,y) = T_y(x, y). In two dimensions, there is a conformal mappingprinciple: "Under the conformal mapping x+iy = f(x^*+iy^*), thefunctions T(x, y) and A(x, y) go into functions associated witha medium having speed C^*,y^*) = C(Re[f), Im[f]/f¹(x^*+iy."Thereis also an unchanged wavefronts principle: "If g is a smoothfunction with g(0) = 0 and g'(0)>0 then T^*(x) = g(T(x) andA^*(x) = A(x)[g'(x)/g'^1/2 are associated with a medium havingspeed C^*(x) = C(x)/g'(T(x))." in two dimensions, alternatingthe application of the last two principles generates a sequenceof media with their associated T(x, y) and A(x, y). Some ofthese can be extended into three dimensions by applying thefirst principle.     

A Multilinear Extension Inequality In Rn

Sharp decay estimates are provided in this paper for sphericalaverages of a certain multilinear extension operator on L² (S^n–1)x ... x L² (S^n–1). 2000 Mathematics Subject Classification42B10.     

On a Conjecture of Erdos on 3-Powerful Numbers

A conjecture of P. Erdös says that the diophantine equationx+y = z has infinitely many solutions with (x,y) = 1 and suchthat if a prime p divides xyz, then p³ divides xyz. In thispaper, we give a proof of this conjecture.     

Damping Oscillatory Integrals with Polynomial Phase and Convolution Operators with the Affine Arclength Measure on Polynomial Curves in Rn

McMichael proved that the convolution with the (euclidean) arclengthmeasure supported on the curve t (t, t², ..., tⁿ), 0 < t< 1, maps L^p(Rⁿ) boundedly into L^p'(Rⁿ) if and only if 2n(n+1)/(n²+n+2) p 2. In proving this, a uniform estimate on damping oscillatoryintegrals with polynomial phase was crucial. In this paper,a remarkably simple proof of the same estimate on oscillatoryintegrals is presented. In addition, it is shown that the convolutionoperator with the affine arclength measure on any polynomialcurve in Rⁿ maps L^p(Rⁿ) boundedly into L^p'(Rⁿ) if p = 2n(n+1)/(n²+n+2).     

An error bound for the modified successive overrelaxation method

In this paper we consider the modified successive overrelaxation(MSOR)methodto appropriate the solution of the linear system D^-1/2 Ax =D^-1/2b, where A is a symmetric, positive definite and consistentlyordered matrix and D is a diagonal matrix with the diagonalidentical to that of A. The main purpose of this paper is to obtain some theoreticalresults, namely a bound for the norm of _n = v –v_n in termsof the norms _n –v_n-1, _n+1 –v_n and their inner product,where v =D^-1/2 x and v_n is the nth iteration vector, obtainedusing the (MSOR)method.     

Connected Lie Groups that Act Freely on a Product of Linear Spheres

In this note we prove that a compact connected Lie group G admitsa free action on some product of linear spheres if and onlyif it is isomorphic to (T^k x SU(2)^l)/Z for some k and l andfor some central elementary abelian 2-subgroup Z with Z SU(2)M^l= 1.     

Doubly stochastic matrix equations

It is shown that for real,m x n matricesA andB the system of matrix equationsAX=B, BY=A is solvable forX andY doubly stochastic if and only ifA=BP for some permutation matrixP. This result is then used to derive other equations and to characterize the Green’s relations on the semigroup Ω _n of alln x n doubly stochastic matrices. The regular matrices in Ω _n are characterized in several ways by use of the Moore-Penrose generalized inverse. It is shown that a regular matrix in Ω _n is orthostochastic and that it is unitarily similar to a diagnonal matrix if and only if it belongs to a subgroup of Ω _n . The paper is concluded with extensions of some of these results to the convex setS _n of alln x n nonnegative matrices having row and column sums at most one. His research was supported by the N. S. F. Grant GP-15943.     

Triangle group representations and constructions of regular maps

A regular map of type {m,n} is a 2-cell embedding of a graphin an orientable surface, with the property that for any twodirected edges e and e' there exists an orientation-preservingautomorphism of the embedding that takes e onto e', and in whichthe face length and the vertex valence are m and n, respectively.Such maps are known to be in a one-to-one correspondence withtorsion-free normal subgroups of the triangle groups T(2,m,n).We first show that some of the known existence results aboutregular maps follow from residual finiteness of triangle groups.With the help of representations of triangle groups in speciallinear groups over algebraic extensions of Z we then constructivelydescribe homomorphisms from T(2,m,n)=y,z|y^m=zⁿ=(yz)²=1 intofinite groups of order at most c^r where c=c(m,n), such thatno non-identity word of length at most r in x,y is mapped ontothe identity. As an application, for any hyperbolic pair {m,n}and any r we construct a finite regular map of type {m,n} ofsize at most C^r, such that every non-contractible closed curveon the supporting surface of the map intersects the embeddedgraph in more than r points. We also show that this result isthe best possible up to determining C=C(m,n). For r>m thegraphs of the above regular maps are arc-transitive, of valencen, and of girth m; moreover, if each prime divisor of m is largerthan 2n then these graphs are non-Cayley. 2000 Mathematics SubjectClassification: 05C10, 05C25, 20F99, 20H25.     

One-Factorizations of Complete Graphs with a Doubly Transitive Automorphism Group

It is shown that a 1-factorization of K_n with a doubly transitiveautomorphism group on vertices is either the affine line-parallelismof AG(d, 2), or one of three ‘sporadic’ exampleswith n = 6, 12 or 28. The full automorphism groups are respectivelyAGL (d, 2) (the holomorph of an elementary abelian group oforder 2^d), PGL(2,5), PSL(2,11) and PL(2,8).     

Small Sets which meet all the n-Term Arithmetic Progressions in the Interval [1, n2]

For any positive integers n and k, let f(n, k) denote the smallestsize of a subset of the integer interval I =[l, n] which meetsall the k-term arithmetic progressions contained in I. We showthat n+(1/2)n^1/2–2 < f(n²,n) , where p is the largest prime n, and for any real number x,[x] is the least integer x.     

Smoothness Properties of the Unit Ball in a JB*-Triple

An element a of norm one in a JB^*-triple A is said to be smoothif there exists a unique element x in the unit ball A₁^* of thedual A^* of A at which a attains its norm, and is said to beFréchet-smooth if, in addition, any sequence (x_n) ofelements in A₁^* for which (x_n(a)) converges to one necessarilyconverges in norm to x. The sequence (a²ⁿ⁺¹) of odd powers ofa converges in the weak^*-topology to a tripotent u(a) in theJBW^*-envelope A^** of A. It is shown that a is smooth if andonly if u(a) is a minimal tripotent in A^** and a is Fréchet-smoothif and only if, in addition, u(a) lies in A.     

Hypersurfaces in a Unit Sphere Sn+1(1) with Constant Scalar Curvature

The paper considers n-dimensional hypersurfaces with constantscalar curvature of a unit sphere S^n–1(1). The hypersurfaceS^k(c₁)xS^n–k(c₂) in a unit sphere Sⁿ⁺¹(1) is characterized,and it is shown that there exist many compact hypersurfaceswith constant scalar curvature in a unit sphere Sⁿ⁺¹(1) whichare not congruent to each other in it. In particular, it isproved that if M is an n-dimensional (n > 3) complete locallyconformally flat hypersurface with constant scalar curvaturen(n–1)r in a unit sphere Sⁿ⁺¹(1), then r > 1–2/n,and (1) when r (n–2)/(n–1), if then M is isometric to S¹xS^n–1(c),where S is the squared norm of the second fundamental form ofM; (2) there are no complete hypersurfaces in Sⁿ⁺¹(1) with constantscalar curvature n(n–1)r and with two distinct principalcurvatures, one of which is simple, such that r = (n–2)/(n–1)and      

On time regularity and related conditions for power-bounded operators

Let T be a bounded linear operator in a complex Banach space.Our main result gives various characterizations of the condition:T is power-bounded and an estimate ||(I – T)Tⁿ || cn^–1/2 holds for all positive integers n. In particular, this conditionholds if and only if T = β S + (1 – β)I, forsome β (0, 1) and some power-bounded operator S; or ifand only if T is power-bounded and the discrete semigroup (Tⁿ)is dominated by the continuous semigroup (e^{– t(I –T)})_{t 0} in a natural sense. As a consequence of our main results,for 1/2 < 1 we characterize the condition that T is power-boundedand ||(I – T)Tⁿ || c n^– for all n, in terms ofestimates on the semigroup e^{–t(I – T)}.     

A Mod Two Analogue of a Conjecture of Cooke

The mod two cohomology of the three connective covering of S³has the form F₂[X_2n] E(Sq¹X_2n) where x_2n is in degree 2n and n = 2. If F denotes the homotopytheoretic fibre of the map S³ B²S¹ of degree 2, then the mod2 cohomology of F is also of the same form for n = 1. Notice(cf. Section 7 of the present paper) that the existence of spaceswhose cohomology has this form for high values of n would immediatelyprovide Arf invariant elements in the stable stem. Hence, itis worthwhile to determine for what values of n the above algebracan be realized as the mod2 cohomology of some space. The purposeof this paper is to construct a further example of a space withsuch a cohomology algebra for n = 4 and to show that no othervalues of n are admissible. More precisely, we prove the following.     

The eigenvalues of limits of radial Toeplitz operators

Let A² be the Bergman space on the unit disk. A bounded operatorS on A² is called radial if Szⁿ = _n zⁿ for all n 0, where _nis a bounded sequence of complex numbers. We characterize theeigenvalues of radial operators that belong to the Toeplitzalgebra.     

On the ideals and singularities of secant varieties of Segre varieties

We find minimal generators for the ideals of secant varietiesof Segre varieties in the cases of _k(¹ x ⁿ x ^m) for all k, n,m, ₂(ⁿ x ^m x ^p x ^r) for all n, m, p, r (GSS conjecture for fourfactors), and ₃(ⁿ x ^m x ^p) for all n, m, p and prove they arenormal with rational singularities in the first case and arithmeticallyCohen–Macaulay in the second two cases.     

Weil Representations of Symplectic Groups Over Rings

We are interested in Weil representations of Sp(2n, R), whereR is the ring Z/p^lZ, p is an odd prime and l is a positive integer,or, more generally, R = O/p^l, where O is the ring of integersof a local field, p is the maximal ideal of O and O/p has oddcharacteristic. One reason for this interest is that a continuousfinite-dimensional complex representation of Sp(2n, O) has tofactor through a representation of Sp(2n, O/p^l) for some l.     

Optimum Runge-Kutta-Fehlberg Methods for Second-order Differential Equations

^* Presently at Deparment of Mathematics, Indian Institute of Technology, Madras, India. The optimum Runge-Kutta method of a particular order is theone whose truncation error is minimum. In this paper, we havederived optimum Runge-Kutta mehtods of 0(h^m+4), 0(h^m+5) and0(h^m+6) for m = 0(1)8, which can be directly used for solvingthe second order differential equation yⁿ = f(x, y, y'). Thesemethods are based on a transformation similar to that of Fehlbergand require two, three and four evaluations of f(x, y, y') respectively,for each step. The numercial solutions of one example obtainedwith these methods are given. It has been assumed that f(x,y, y')is sufficiently differentiable in the entire region ofintegration.     

Spines and Embeddings of n-Manifolds

Every compact, connected PL manifold Mⁿ, with MⁿØ, collapsesto a codimension-one subpolyhedron Q^n–1, called a spineof Mⁿ. The purpose of this paper is to prove that, if Q^n–1is appropriately chosen, one can reconstruct Mⁿ from Q^n–1,after taking the Cartesian product with an interval I=[0, 1].     

The quaternion group as a subgroup of the sphere braid groups

Let n 3. We prove that the quaternion group of order 8 is realisedas a subgroup of the sphere braid group B_n(²) if and only ifn is even. If n is divisible by 4, then the commutator subgroupof B_n(²) contains such a subgroup. Further, for all n 3, B_n(²)contains a subgroup isomorphic to the dicyclic group of order4n.