Multistep approximation of linear sectorial evolution equations

This paper concerns the linear multistep approximation of alinear sectorial evolution equation u_t = Au on a complex Banachspace X. Given a strictly A()-stable q-step method of orderp whose stability region includes a sectorial region containingthe spectrum of the operator A, the corresponding evolutionsemigroup for the method is Cⁿ(hA), n 0, defined on X^q, whereC(z) L (C^q) denotes the one-step map associated with the method.It is shown that for appropriately chosen V, Y: C C^q, basedon the principal right and left eigenvectors of C(z), Cⁿ(hA)approximates the semigroup V(hA)e^nhAY^H(hA) with optimal orderp.     

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.     

Some New and Old Asymptotic Representations of the Jost Solution and the Weyl m-Function For Schrodinger Operators on the Line

For the general one-dimensional Schrödinger operator –d²/dx²+q(x) with real q L₁(R), this paper presents a new series representationof the Jost solution which, in turn, implies a new asymptoticrepresentation of the Weyl m-function for locally summable q.This representation is then applied to smooth potentials q toobtain Weyl m-function power asymptotics. The condition q^(N) L₁(x₀, x₀ + ), for N N₀, allows one to derive the (N + 1)term for almost all x [x₀, x₀ + ), thereby refining a relevantresult by Danielyan, Levitan and Simon. 2000 Mathematics SubjectClassification 34E05, 34L40 (primary), 34B20, 34L25 (secondary).     

A Matrix Setting for the q-Schur Algebra

The Schur algebra S(n, r) has a basis (described in [6, 2.3])consisting of certain elements _i,j, where i, jI(n, r), the setof all ordered r-tuples of elements from the set n={1, 2, ...,n}. The multiplication of two such basis elements is given bya formula known as Schur's product rule. In recent years, aq-analogue S_q(n, r) of the Schur algebra has been investigatedby a number of authors, particularly Dipper and James [3, 4].The main result of the present paper, Theorem 3.6, shows howto embed the q-Schur algebra in the rth tensor power T^r(M_n)of the nxn matrix ring. This embedding allows products in theq-Schur algebra to be computed in a straightforward manner,and gives a method for generalising results on S(n, r) to S_q(n,r). In particular we shall make use of this embedding in subsequentwork to prove a straightening formula in S_q(n, r) which generalisesthe straightening formula for codeterminants due to Woodcock[12]. We shall be working mainly with three types of algebra: thequantized enveloping algebra U(gl_n) corresponding to the Liealgebra gl_n, the q-Schur algebra S_q(n, r), and the Hecke algebra,H(A_r–1). It is often convenient, in the case of the q-Schuralgebra and the Hecke algebra, to introduce a square root ofthe usual parameter q which will be denoted by v, as in [5].This corresponds to the parameter v in U(gl_n). We shall denotethis ‘extended’ version of the q-Schur algebra byS_v(n, r), and we shall usually refer to it as the v-Schur algebra.All three algebras are associative and have multiplicative identities,and the base field will be the field of rational functions,Q(v), unless otherwise stated. The symbols n and r shall bereserved for the integers given in the definitions of thesethree algebras.     

Binary Quadratic Forms with Large Discriminants and Sums of Two Squareful Numbers II

Let F = (F₁, ..., F_m) be an m-tuple of primitive positive binaryquadratic forms and let U_F(x) be the number of integers notexceeding x that can be represented simultaneously by all theforms F_j, j = 1, ... , m. Sharp upper and lower bounds for U_F(x)are given uniformly in the discriminants of the quadratic forms. As an application, a problem of Erds is considered. Let V(x)be the number of integers not exceeding x that are representableas a sum of two squareful numbers. Then V(x) = x(log x)^–+o(1)with = 1 – 2^–1/3 = 0.206....     

Essential norms and localization moduli of Sobolev embeddings for general domains

We introduce new measures of non-compactness for the embeddingoperator E_p,q():L_p¹() L_q() and describe their relations withthe essential norm of E_{p, q}(), ‘local’ isoperimetricand isocapacitary constants. An explicit formula for the essentialnorm of E_{p, q}() is obtained for domains with a power cusp onthe boundary and bounded C¹ domains. The Neumann problem fora particular Schrödinger operator is discussed on domainswith a power cusp.     

On the L1 Mean of the Exponential Sum Formed with the Mobius Function

In this paper we study the L¹ mean (1) of the exponential sum M()=_nXµ(n)e(n), where µ(n)is the Möbius function and e(x)=e^2ix. From the Cauchy–Schwarzinequality and Parseval's identity, we have , (2) and it is an interesting problem to investigate whether (2)reflects the true order of magnitude of (1).     

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.     

Recurrence and asymptotics for orthonormal rational functions on an interval

^{Let µ be a positive bounded Borel measure on a subsetI of the real line and = {₁, ..., _n} a sequence of arbitrary ‘complex’poles outside I. Suppose {₁, ..., _n} is the sequence of rationalfunctions with poles in orthonormal on I with respect to µ. First, we are concernedwith reducing the number of different coefficients in the three-termrecurrence relation satisfied by these orthonormal rationalfunctions. Next, we consider the case in which I = [–1, 1] and µ satisfies the Erdos–Turán conditionµ' > 0 a.e. on I (where µ' is the Radon–Nikodymderivative of the measure µ with respect to the Lebesguemeasure) to discuss the convergence of _n+1(x)/_n(x) as n tendsto infinity and to derive asymptotic formulas for the recurrencecoefficients in the three-term recurrence relation. Finally,we give a strong convergence result for _n(x) under the morerestrictive condition that µ satisfies the Szeg condition(1 – x2)–1/2 log µ'(x) L1([– 1, 1]).}     

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.     

De Rham Cohomology and Hodge Decomposition For Quantum Groups

Let be one of the N²-dimensionalbicovariant first order differential calculi for the quantumgroups GL_q(N), SL_q(N), SO_q(N), or Sp_q(N), where q is a transcendentalcomplex number and z is a regular parameter. It is shown thatthe de Rham cohomology of Woronowicz' external algebra coincides with the de Rham cohomologiesof its left-coinvariant, its right-coinvariant and its (two-sided)coinvariant subcomplexes. In the cases GL_q(N) and SL_q(N) thecohomology ring is isomorphic to the coinvariant external algebra and to the vector space of harmonic forms. We prove a Hodge decomposition theorem in thesecases. The main technical tool is the spectral decompositionof the quantum Laplace-Beltrami operator. 2000 MathematicalSubject Classification: 46L87, 58A12, 81R50.     

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      

Controllability of a system of parabolic equations with non-diagonal diffusion matrix

In this paper we give a necessary and sufficient algebraic conditionfor the approximate controllability of the following systemof parabolic equations with Dirichlet boundary condition: {z_t = D z + b₁(x)u₁ + ··· + b_m(x)u_m, t 0, z ⁿ, z = 0, on where is a sufficiently smooth bounded domain in ^N, b_i L²(;ⁿ), the control functions u_i L²(0, t₁; ); i = 1, 2, ..., mand D is an n x n non-diagonal matrix whose eigenvalues aresemi-simple with positive real part. This algebraic conditionis checkable since it is given in terms of the n_j x m matricesDP_j and P_jB, i.e. Rank [P_jBDP_jBD²P_jB··· D^n_j–1 P_jB]= n_j, where P_jBu = P_jb₁u₁ + ··· + P_jb_mu_m. Finally,this result can be applied to those systems of partial differentialequations that can be rewritten as a diffusion system (see deOliveira, 1998).     

Long-Time Behavior of Solutions of the Fast Diffusion Equations with Critical Absorption Terms

This paper is devoted to the long-time behavior of solutionsto the Cauchy problem of the porous medium equation u_t = (u^m)– u^p in Rⁿ x (0,) with (1 – 2/n)₊ < m < 1and the critical exponent p = m + 2/n. For the strictly positiveinitial data u(x,0) = O(1 + |x|)^–k with n + mn(2 –n + nm)/(2[2 – m + mn(1 – m)]) k < 2/(1 –m), we prove that the solution of the above Cauchy problem convergesto a fundamental solution of u_t = (u^m) with an additional logarithmicanomalous decay exponent in time as t .     

The tower of K-theory of truncated polynomial algebras

Let A be a regular noetherian F_p-algebra. The relative K-groupsK_q(A[x]/(x^m),(x)) and the Nil-groups Nil_q(A[x]/(x^m)) were evaluatedby the author and Ib Madsen in terms of the big de Rham–Wittgroups W_rA^q of the ring A. In this paper, we evaluate the mapsof relative K-groups and Nil-groups induced by the canonicalprojection f: A[x]/(x^m) A[x]/(xⁿ). The result depends stronglyon the prime p. It generalizes earlier work by Stienstra onthe groups in degrees 2 and 3. Received February 28, 2007.     

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.     

A Lie Theoretic Approach to Thompson's Theorems on Singular Values-Diagonal Elements and Some Related Results

Thompson's famous theorems on singular values–diagonalelements of the orbit of an nxn matrix A under the action (1)U(n) U(n) where A is complex, (2) SO(n) SO(n), where A isreal, (3) O(n) O(n) where A is real are fully examined. Coupledwith Kostant's result, the real semi-simple Lie algebra so_n,n yields (2) and hence (3) and the sufficient part (the hardpart) of (1). In other words, the curious subtracted term(s)are well explained. Although the diagonal elements correspondingto (1) do not form a convex set in Cⁿ, the projection of thediagonal elements into Rⁿ (or iRⁿ) is convex and the characterizationof the projection is related to weak majorization. An elementaryproof is given for this hidden convexity result. Equivalentstatements in terms of the Hadamard product are also given.The real simple Lie algebra su_{n, n} shows that such a convexityresult fits into the framework of Kostant's result. Convexityproperties and torus relations are studied. Thompson's resultson the convex hull of matrices (complex or real) with prescribedsingular values, as well as Hermitian matrices (real symmetricmatrices) with prescribed eigenvalues, are generalized in thecontext of Lie theory. Also considered are the real simple Liealgebras so_{p, q} and so_{p, q}, p < q, which yield the rectangularcases. It is proved that the real part and the imaginary partof the diagonal elements of complex symmetric matrices withprescribed singular values are identical to a convex set inRⁿ and the characterization is related to weak majorization.The convex hull of complex symmetric matrices and the convexhull of complex skew symmetric matrices with prescribed singularvalues are given. Some questions are asked.     

The determination of a coefficient in a parabolic equation from input sources

The authors consider the question of recovering the coefficientq from the equation u_t – u_xx + q(x)u = f_j(x) with homogeneousinitial and boundary conditions. The nonhomogeneous source terms form a basis for L²(0,1).It is proved that a unique determination is possible from datameasurements consisting of either the flux at one end of thebar or the net flux leaving the bar, taken at a single fixedtime for each input source. An algorithm that allows efficientnumerical reconstruction of q(x) from finite data is given.     

Interpolating Blaschke Products and Factorization Theorems

Let M(H) be the maximal ideal space of H the Banach algebraof bounded analytic functions on the open unit disk. Let G bethe set of nontrivial points in M(H). By Hoffman's work, G hasdeep connections with the zero sets of interpolating Blaschkeproducts. It is proved that for a closed -separated subset Eof M(H) with E G, there exists an interpolating Blaschke productwhose zero set contains E. This is a generalization of Lingenberg'stheorem. Let f be a continuous function on M(H). Suppose thatf is analytic on a nontrivial Gleason part P(x), f(x) = 0, andf 0 on P(x). It is proved that there is an interpolating Blaschkeproduct b with zeros {z_n}_n such that b(x) = 0 and f(z_n) = 0for every n. This fact can be used for factorization theoremsin Douglas algebras and in algebras of functions analytic onGleason parts.