Shapiro's Cyclic Sum

Shapiro's cyclic sum is defined by , If K is the cone in Rⁿ of points withnon-negative coordinates, it is shown that the minimum of Ein K is a fixed point of T², where T is the non-linear operatordefined by (Tx)_i = x_n–i+1/(x_n–i+2 + x_n–i+3)²for i = 1,2,...,n. It is conjectured that Tx = S^kx, where Sis the shift operator in Rⁿ, and a proof is given under someadditional hypotheses. One of the consequences is a simple proofthat at the minimum point, a_i(x) = a_n–i+1–k(x) fori = 1,2,...,n.     

Generalized Alternating Polynomials, some Properties and Numerical Applications

In the present paper we introduce the generalized alternatingpolynomials , the coefficients of which are defined together with the parameter W_n by the linear system where T_n(x) = cos (n arc cos x), is a set of n+2 distinct points in the interval [–1, 1], and fis a continuous function on [–1, 1], For the set of nodesx_k = cos [k/(n+1)] the w_n-polynomials coincide with the polynomialsof equiamplitude alternation introduced by de La Vall?e-Poussinand discussed in the literature earlier (Eterman, Malozemov,Meinardus, Cheney & Rivlin, Phillips & Taylor, Brutman,and others). It is shown that the generalized alternating polynomials arerelated to the polynomials of interpolation through the Lanczoseconomization process. Some approximation properties of w_n -polynomialsand W_n -parameters are studied. The application of w_n -polynomialsto function approximation and to the estimate of remainder termsfor quadrature formulas is discussed.     

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      

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 new transform method for evolution partial differential equations

We introduce a new transform method for solving initial-boundary-valueproblems for linear evolution partial differential equationswith spatial derivatives of arbitrary order. This method isillustrated by solving several such problems on the half-line{t > 0, 0 < x < }, and on the quarter-plane {t >0, 0 < x_j < , j = 1, 2}. For equations in one space dimensionthis method constructs q(x, t) as an integral in the complexk-plane involving an x-transform of the initial condition anda t-transform of the boundary conditions. For equations in twospace dimensions it constructs q(x₁, x₂, t) as an integral inthe complex (k₁, k₂)-planes involving an (x₁, x₂)-transformof the initial condition, an (x₂, t)-transform of the boundaryconditions at x₁ = 0, and an (x₁, t)-transform of the boundaryconditions at x₂ = 0. This method is simple to implement andyet it yields integral representations which are particularlyconvenient for computing the long time asymptotics of the solution.     

On the Optimum Criterion of Polynomial Stability

The purpose of this note is to answer the question raised byNie & Xie (1987). Let f(x)=a₀xⁿ+a₁x^n–1+...+a_n be apositive-coefficient polynomial. The numbers ₁=a_i-1a_i+2/a_ia_i+1(i=1, ..., n–2) are called determining coefficients. Theoptimum criterion problem was posed as follows: for n3, findthe maximal number (n) such that the polynomial f(x) is stableif _i < (n) (1in–2). For n6, we show that (n)=ß,where ß is the unique real root of the equation x(x+1)²=1.     

Rado Partition Theorem for Random Subsets of Integers

For an l x k matrix A = (a_ij) of integers, denote by L(A) thesystem of homogenous linear equations a_i1x₁ + ... + a_ikx_k =0, 1 i l. We say that A is density regular if every subsetof N with positive density, contains a solution to L(A). Fora density regular l x k matrix A, an integer r and a set ofintegers F, we write if for any partition F = F₁ ... F_r there exists i {1, 2,..., r} and a column vector x such that Ax = 0 and all entriesof x belong to F_i. Let [n]_N be a random N-element subset of{1, 2, ..., n} chosen uniformly from among all such subsets.In this paper we determine for every density regular matrixA a parameter = (A) such that lim_n P([n]_N (A)_r)=0 if N =O(n) and 1 if N = (n). 1991 Mathematics Subject Classification:05D10, 11B25, 60C05     

Fast Solution of Vandermonde-Like Systems Involving Orthogonal Polynomials

Consider the (n + 1) ? (n + 1) Vandermonde-like matrix P=[p_i-1(_j-1)],where the polynomials p_o(x), ..., p_n(x) satisfy a three-termrecurrence relation. We develop algorithms for solving the primaland dual systems, Px = b and P^Ta = f respectively, in O(n²)arithmetic operations and O(n) elements of storage. These algorithmsgeneralize those of Bj?rck & Pereyra which apply to themonomial case p_i(x). When the p_i(x) are the Chebyshev polynomials,the algorithms are shown to be numerically unstable. However,it is found empirically that the addition of just one step ofiterative refinement is, in single precision, enough to makethe algorithms numerically stable.     

Random Sequences Interpolating with Probability One

Let {_n} be a sequence of independent random variables uniformlydistributed on [0, 2], and let {r_n} be a sequence of (deterministic)radii in [0, 1). Form points of the unit disc putting z_n = r_ne_n.We characterize those sequences {r_n} for which {z_n} is an interpolatingsequence with probability one.     

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.     

On the nth Quantum Derivative

The nth quantum derivative D_nf(x) of the real-valued functionf is defined for each real non-zero x as where is the q-binomial coefficient.If the nth Peano derivative exists at x, which is to say thatif f can be approximated by an nth degree polynomial at thepoint x, then it is not hard to see that D_nf(x) must also existat that point. Consideration of the function |1–x| atx = 1 shows that the second quantum derivative is more generalthan the second Peano derivative. However, it can be shown thatthe existence of the nth quantum derivative at each point ofa set necessarily implies the existence of the nth Peano derivativeat almost every point of that set.     

A Numerical Method for the Solution of the Double Eigenvalue Problem

A generalization of the Rayleigh quotient iterative method,called the Minimum Residual Quotient Iteration (MRQI), is derivedfor the numerical solution of the 2-parameter eigenvalue problem;i.e. to find scalars µ and a corresponding vector x satisfyingthe following equations, Ax = B₁x + µB₂x, ||x|| = 1, f(x) = 0, where A and B are nxn real matrices, ||.|| denotes the l₂ normand f is a real functional. The method is applied to doubleeigenvalue problems for ordinary differential equations andcomputational results are presented.     

BENFORD'S LAW FOR THE 3x + 1 FUNCTION

Benford's law (to base B) for an infinite sequence {x_k : k 1} of positive quantities x_k is the assertion that {log_B x_k: k 1} is uniformly distributed (mod 1). The 3x + 1 functionT(n) is given by T(n) = (3n + 1)/2 if n is odd, and T(n) = n/2if n is even. This paper studies the initial iterates x_k = T^(k)(x₀)for 1 k N of the 3x + 1 function, where N is fixed. It showsthat for most initial values x₀, such sequences approximatelysatisfy Benford's law, in the sense that the discrepancy ofthe finite sequence {log_B x_k : 1 k N} is small.     

Multistep Discretization of Linear Hyperbolic Equations

Order and stability of multistep finite-difference discretizationsof the first-order linear hyperbolic equation u₁ = a(x)u_x areconsidered. We prove that if a stable method uses s upwind andrdownwind points and the coefficients depend only on the Courantnumber and on a(x) and its derivatives at the underlying gridpoint, then the order may not exceed r + s. This bound on orderis exactly half the bound of Strang & Iserles (1983) forconstant a. Furthermore, we prove that if r = s and a(x) isboth uniformly bounded and uniformly positive for x R thenthe new order barrier is attainable for every s 1.     

A Krylov subspace algorithm for multiquadric interpolation in many dimensions

We consider the version of multiquadric interpolation wherethe interpolation conditions are the equations s(x_i) = f_i, i= 1,2,..., n, and where the interpolant has the form s(x) =_j=1ⁿ _j (||x – x_j ||² + c²)^1/2 + x R^d, subject to theconstraint _j=1ⁿ _j = 0. The points x_i R^d, the right-hand sidesf_i, i = 1,2,...,n, and the constant c are data. The equationsand the constraint define the parameters _j, j = 1,2,...,n, and. The resultant approximation s f is useful in many applications,but the calculation of the parameters by direct methods requiresO (n³) operations, and n may be large. Therefore iterative proceduresfor this calculation have been studied at Cambridge since 1993,the main task of each iteration being the computation of s(x_i),i = 1,2,...,n, for trial values of the required parameters.These procedures are based on approximations to Lagrange functions,and often they perform very well. For example, ten iterationsusually provide enough accuracy in the case d = 2 and c = 0,for general positions of the data points, but the efficiencydeteriorates if d and c are increased. Convergence can be guaranteedby the inclusion of a Krylov subspace technique that employsthe native semi-norm of multiquadric functions. An algorithmof this kind is specified, its convergence is proved, and carefulattention is given to the choice of the operator that definesthe Krylov subspace, which is analogous to pre-conditioningin the conjugate gradient method. Finally, some numerical resultsare presented and discussed, for values of d and n from theintervals [2,40] and [200,10 000], respectively.     

On the Genus of a Finite Classical Group

Let G be a finite group acting faithfully and transitively ona set of size m, and let E = {x₁, ..., x_k} be a generatingset for G with x₁x₂...x_k = 1. If x G has cycles of length r₁,..., r_l in its action on , define . Then the genus g = g(G, , E) is defined by 1991 Mathematics Subject Classification 20B25, 20G40,30F99.     

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^*.     

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.     

Fractional decompositions of dense hypergraphs

A seminal result of Rödl (the Rödl nibble) assertsthat the edges of the complete r-uniform hypergraph can be packed, almost completely, with copiesof , where k is fixed.We prove that the same result holds in a dense hypergraph setting.It is shown that for every r-uniform hypergraph H₀, there existsa constant = (H₀) < 1 such that every r-uniform hypergraphH in which every (r – 1)-set is contained in at least n edges has an H₀-packing that covers |E(H)|(1 – o_n(1))edges. Our method of proof uses fractional decompositions andmakes extensive use of probabilistic arguments and additionalcombinatorial ideas.     

Factor Algebras of Free Algebras: On a Problem of G. Bergman

Let A_n = K x₁,...,x_n be a free associative algebra over a fieldK. In this paper, examples are given of elements u A_n, n 3,such that the factor algebra of A_n over the ideal generatedby u is isomorphic to A_n–1, and yet u is not a primitiveelement of A_n (that is, it cannot be taken to x₁ by an automorphismof A_n). If the characteristic of the ground field K is 0, thisyields a negative answer to a question of G. Bergman. On theother hand, by a result of Drensky and Yu, there is no suchexample for n = 2. It should be noted that a similar questionfor polynomial algebras, known as the embedding conjecture ofAbhyankar and Sathaye, is a major open problem in affine algebraicgeometry. 2000 Mathematics Subject Classification 16S10, 16W20(primary); 14A05, 13B25 (secondary).