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).     

A note on S(t) and the zeros of the Riemann zeta-function

Let S(t) denote the argument of the Riemann zeta-function atthe point 1/2 + it. Assuming the Riemann hypothesis, we sharpenthe constant in the best currently known bounds for S(t) andfor the change of S(t) in intervals. We then deduce estimatesfor the largest multiplicity of a zero of the zeta-function,and for the largest gap between the zeros.     

On the dynamics of a discretized neutral equation

In this paper, we investigate the stability properties of numericalmethods for solving the differential equation of the neutraltype y'(t) = Ay(t) + By(qt) + Cy'(pt), y(0) = 1, where p, q (0, 1), A, B, C and A 0. Sufficient conditions for the asymptoticstability of a discretized model of this equation are givenwhen p = q = L^–1, where L 2 is an integer. These coincidewith conditions for asymptotic stability of the original equation,as reported in (Iserles, 1991), except that the step-lengthneed be restricted and, at the limit, asymptotic stability isretained only if |C| < 1. Moreover, for a particular choiceof A these conditions are also shown to be necessary.     

On the Compactness of the Nonlinear Evolution Operator in a Banach Space

Let X be a real Banach space and let A(t): X 2^x be dissipativefor all t(0, T). Assume that {A(t)} generates an evolution operatorU(t, s) of type (D, , f). Necessary and sufficient conditionsare given for the compactness of U(t, s) for 0 s < t <T.     

Classification Des Ideaux a Droite De A1(k)

Les études récentes sur les idéaux àdroite de A₁(k), la première algèbre de Weyl surun corps algébriquement clos et de caractéristiquenulle k, nous montrent que : pour tout idéal I 0 àdroite de A₁(k), il existe x Q = frac(A₁(k)), et V V telsque : I = xD(R, V) o V est l'ensemble des sous-espaces primairementdécomposables de k[t] = R, et D(R, V), l'idéalà droite {d A₁(k/d(R V}. Dans cet article nous montreronsprincipalement que: pour tout 0 I idéal à droitede A₁(k, !n N, (x, ) Q^* x Aut_k(A₁(k)) : I = x(D(R, O(X_n))),où X_n est la courbe d'algèbre des fonctions régulières: O(X_n = k+tⁿ⁺¹k[t]. La forme des idéaux décriteci-dessus permet de voir dans une hypothèse de Letzteret Makar-Limanov, pour deux courbes algébriques affinesX et X' on a : D(XD(X') co dim D(X = co dim D(X'). Recent studies on right ideals of the first Weyl algebra A₁(k)over an algebraic closed field k with characteristic zero showthat: for each right ideal I 0 of A₁(k), there exist x Q =fracA₁(k)) and a primary decomposable sub-space V of k[t] suchthat I=xD(R,V), where D(R,V) : = {d A₁(k)/d(R) V} is a rightideal of A₁(k). In this paper, we show that for all right idealsI 0 of A₁(k), !n N, (x, ) Q^* x Aut_k(A₁(k)) : I = x(D(R, O(X_n))),where X_n denotes the affine algebraic curve with ring of regularfunctions O(X_n=k+tⁿ⁺¹k[t]. With ideals as described above, onecan easily see, under a hypothesis given by Letzter and Makar-Limanov,that for two affine algebraic curves X and X', D(X)D(X') codim D(X) = co dim D(X'). 2000 Mathematics Subject Classification16S32.     

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 .     

Semidiscrete finite element Galerkin approximations to the equations of motion arising in the Oldroyd model

In this paper, a semidiscrete finite element Galerkin methodfor the equations of motion arising in the 2D Oldroyd modelof viscoelastic fluids with zero forcing function is analysed.Some new a priori bounds for the exact solutions are derivedunder realistically assumed conditions on the data. Moreover,the long-time behaviour of the solution is established. By introducinga Stokes–Volterra projection, optimal error bounds forthe velocity in the L(L²) as well as in the L(H¹)-norms andfor the pressure in the L(L²)-norm are derived which are validuniformly in time t > 0.     

Optimal Regularity and Fredholm Properties of Abstract Parabolic Operators in Lp Spaces on the Real Line

We study the operator Lu(t):= u'(t) – A(t) u(t) on L^p(R; X) for sectorial operators A(t), with t R, on a Banachspace X that are asymptotically hyperbolic, satisfy the Acquistapace–Terreniconditions, and have the property of maximal L^p-regularity.We establish optimal regularity on the time interval R showingthat L is closed on its minimal domain. We further give conditionsfor ensuring that L is a semi-Fredholm operator. The Fredholmproperty is shown to persist under A(t)-bounded perturbations,provided they are compact or have small A(t)-bounds. We applyour results to parabolic systems and to generalized Ornstein–Uhlenbeckoperators. 2000 Mathematics Subject Classification 35K20, 35K90,47A53.     

A division theorem for real analytic functions

We characterize those homogeneous polynomials P [z₁, ... ,z_d] for which the principal ideal (P) = P · A(^d) is complementedin A(^d) or, equivalently, those which admit a continuous lineardivision operator. The condition is the same as that which characterizes,among the homogeneous polynomials, those which are nonellipticand for which P(D) is surjective in A(^d), and those for whichP(D) admits a continuous linear right inverse in C(^d). It dependsonly on the type of real singularities.     

Numerical Methods for y' =f(x, y) via Rational Approximations for the Cosine

To investigate stability and phase lag, a numerical method isapplied to the test equation y' = –²y. Frequently, thecharacteristic equation of the resulting recurrence relationhas the form ²– 2R_nm(v²) + 1 = 0, where v = h, with hthe steplength, and R_nm(v²) is a rational approximation forcos v. In this paper, properties of such approximations areused to provide a general framework for the study of stabilityintervals and orders of dispersion of a variety of one- andtwo-step methods. Upper bounds on the intervals of periodicityof explicit methods with maximum order of dispersion are established.It is shown that the order of dispersion of a P-stable method,for given n and m, cannot exceed 2m; a consequence is that,of the Pad? approximants for cos v, only the [0/2m] approximantshave modulus less than unity for all v² >0. A complete characterizationof P-stable methods of fourth order corresponding to the rationalapproximation R₂₂(v²) is followed by several results for methodswhich have finite intervals of periodicity; in particular, weidentify methods which have order of dispersion 6 or 8 withlarge intervals of periodicity. There is also a detailed discussionof P-stable methods of sixth order corresponding to the rationalapproximation R₃₃(v²).     

A Bound for Solutions of a Fourth-order Dynamical System

We consider the homogeneous linearized equation of yawing motionof a spinning projectile ⁿ+A)s)' + B(s) = 0, where A and B are complex functions of the real independentvariable s. It is shown that where K is a positive constant and v(s) is a real function ofs.     

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 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.     

Implicit finite difference schemes for a linear model of well-reservoir coupling

Standard reservoir models usually consider wells as Dirac measuresover an interval length. Moreover, the well-reservoir couplingis taken into account under quite simplified assumptions. Mostrecently, however, attention has been drawn to the fact thatin some situations, such as those related to non-vertical wells,these simplifications do not allow us to model some relevantmechanisms of the coupled flow. Therefore, more complex alternativemodels have been proposed recently in the oil reservoir simulationliterature. A linearized version for the well-reservoir couplingcan be written, in an appropriate functional setting, in theform U'(t) + AU(t) = F(t). In this work we discuss implicitin time discretizations of this equation, of the form { Uⁿ⁺¹_h - Uⁿ_h÷+A_hUⁿ⁺¹_h = Fⁿ⁼¹_h, U⁰_h=U_0.h We propose two different approximations, corresponding to first-and second-order spatial truncation errors, and we establishthe convergence of both approximations.     

Witt Groups of the Punctured Spectrum of a 3-Dimensional Regular Local Ring and a Purity Theorem

Let A be a regular local ring with quotient field K. Assumethat 2 is invertible in A. Let W(A)W(K) be the homomorphisminduced by the inclusion AK, where W( ) denotes the Witt groupof quadratic forms. If dim A4, it is known that this map isinjective [6, 7]. A natural question is to characterize theimage of W(A) in W(K). Let Spec¹(A) be the set of prime idealsof A of height 1. For PSpec¹(A), let _P be a parameter of thediscrete valuation ring A_P and k(P) = A_P/PA_P. For this choiceof a parameter _P, one has the second residue homomorphism _P:W(K)W(k(P))[9, p. 209]. Though the homomorphism _P depends on the choiceof the parameter _P, its kernel and cokernel do not. We havea homomorphism A part of the so-called Gersten conjecture is the followingquestion on ‘purity’. Is the sequence exact? This question has an affirmative answer for dim(A)2 [1;3, p. 277]. There have been speculations by Pardon and Barge-Sansuc-Vogelon the question of purity. However, in the literature, thereis no proof for purity even for dim(A) = 3. One of the consequencesof the main result of this paper is an affirmative answer tothe purity question for dim(A) = 3. We briefly outline our main result.     

On the Error Term for the Fourth Moment of the Riemann Zeta-Function

Let E₂(T) denote the error term in the asymptotic formula for^T₀|(+it)|⁴dt. It is proved that there exist constants A>0,B>1 such that for TT₀>0 every interval [T, BT] containspoints T₁, T₂ for which and that ^T₀|E₂(t)|^adt>>T^1+(a/2) for any fixed a1. Theseresults complement earlier results of Motohashi and Ivi that^T₀E₂(t)dt<<T^3/2 and that ^T₀E²₂(t)dt<<T²log^CT forsome C>0. Omega-results analogous to the above ones are obtainedalso for the error term in the asymptotic formula for the Laplacetransform of |(+it)|⁴.     

The Banach Space B(l2) is Primary

We prove that if A is an injective operator system on l² andP is a completely bounded projection on A then either PA or(I–P)A is completely boundedly isomorphic to A. We alsoprove that if B(l²) is linearly homeomorphic to X Y then eitherX or Y is linearly homeomorphic to B(l²). Current address: Merton College, Oxford 0X1 4JD     

The Existence of Periodic and Subharmonic Solutions of Subquadratic Second Order Difference Equations

By critical point theory, a new approach is provided to studythe existence of periodic and subharmonic solutions of the secondorder difference equation where f C(R x R^m, R^m), f(t+M,z)+f(t,z) for any (t, z)R x R^mand M is a positive integer. This is probably the first timecritical point theory has been applied to deal with the existenceof periodic solutions of difference systems.     

Canonical curves in P3

Let Hilb^6t–3(P³) be the Hilbert scheme of closed 1-dimensionalsubschemes of degree 6 and arithmetic genus 4 in P³. Let H bethe component of Hilb^6t–3(P³) whose generic point correspondsto a canonical curve, that is, a complete intersection of aquadric and a cubic surface in P³. Let F be the vector spaceof linear forms in the variables z₁, z₂, z₃, z₄. Denote by F_dthe vector space of homogeneous forms of degree d. Set X = (f₂,f₃)where f₂ P(F₂) is a quadric surface, and f₃ P(F₃/f₂ ·F) is a cubic modulo f₂. Wehave a rational map, : X ... Hdefined by (f₂,f₃) f₂ f₃. It fails to be regular along thelocus where f₂ and f₃ acquire a common linear component. Ourmain result gives an explicit resolution of the indeterminaciesof as well as of the singularities of H. 2000 Mathematical Subject Classification: 14C05, 14N05, 14N10,14N15.     

A new methodology for the stability analysis of pairwise triangularizable and related switching systems

We consider the asymptotic stability of the time-varying dynamicsystem : = A(t)x, A(t) R^{n x n}, A(t) A= {A₁, ..., A_m}, where A_i is Hurwitz and where a set of non-singularmatrices T_{i j} exist such that any pair of matrices {T_{i j} A_iT_{i j}^–1, T_{i j} A_j T_{i j}^–1}, i, j {1, ..., m}, areupper triangular. Switching systems of this form are referredto as pairwise triangularizable switching systems. It can beestablished that (a) pairwise triangularizability is not sufficientto guarantee the existence of a common quadratic Lyapunov functionfor the linear time-invariant dynamic systems A_i : = A_i x; (b) additional conditions can be specified which guaranteeasymptotic stability of the switching system . In this paperwe also show that pairwise triangularizability is not even sufficientto guarantee asymptotic stability of the switching system .We also show that the method of proof of stability in (b), whichdoes not assume the existence of a common quadratic Lyapunovfunction, can be used to prove the asymptotic stability of moregeneral switching systems (systems that are not pairwise triangularizable).Finally, we show that our results can be used as the basis forthe design of practical control systems; namely, for the designof an automobile speed switched controller with guaranteed stabilityproperties.