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.     

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....     

A Nonoscillation Theorem for Second-Order Nonlinear Differential Equations with Decaying Coefficients

The purpose of this paper is to give sufficient conditions forall nontrivial solutions of the nonlinear differential equationx ' a(t)g(x) = 0 to be nonoscillatory. Here, g(x) satisfiesthe sign condition xg(x) > 0 if x 0, but is not assumedto be monotone increasing. This differential equation includesthe generalized Emden–Fowler equation as a special case.Our main result extends some nonoscillation theorems for thegeneralized Emden–Fowler equation. Proof is given by meansof some Liapunov functions and phase-plane analysis.     

Volterra Convolution Operators with Values in Rearrangement Invariant Spaces

The Volterra convolution operator Vf(x) = ^x₀(x–y)f(y)dy,where (·) is a non-negative non-decreasing integrablekernel on [0, 1], is considered. Under certain conditions onthe kernel , the maximal Banach function space on [0, 1] onwhich the Volterra operator is a continuous linear operatorwith values in a given rearrangement invariant function spaceon [0, 1] is identified in terms of interpolation spaces. Thecompactness of the operator on this space is studied.     

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.     

Best Approximation in an Asymmetrically Weighted L1 Measure

Let f(x) be a given, real-valued, continuous function definedon an interval [a,b]of the real line. Given a set of m real-valued,continuous functions _j(x) defined on [a,b], a linear approximatingfunction can be formed with any real setA = {a₁, a₂,..., a_m}. We present results for determining A sothat F(A, x) is a best approximation to(x) when the measureof goodness of approximation is a weighted sum of |F(A, x)–f(x)|,the weights being positive constants, w, when F(A, x) f(x)and w2 otherwise (when w, = w2 = 1, the measure is the L₁, norm).The results are derived from a linear programming formulationof the problem. In particular, we give a theorem which shows when such bestapproximations interpolate the function at fixed ordinates whichare independent of f(x). We show how the fixed points can becalculated and we present numerical results to indicate thatthe theorem is quite robust.     

Finiteness of integrals of functions of Levy processes

We prove necessary and sufficient conditions for the almostsure convergence of the integrals

and thus of ,where M_t = sup{|X_s|: s t} is the two-sided maximum processcorresponding to a Lévy process (X_t)_{t 0}, a(·)is a non-decreasing function on [0, ) with a(0) = 0, g(·)is a positive non-increasing function on (0, ), possibly withg(0 + ) = , and f(·) is a positive non-decreasing functionon [0, ) with f(0) = 0. The conditions are expressed in termsof the canonical measure, (·), of the process X_t. Thespecial case when a(x) = 0, f(x) = x and g(·) is equivalentto the tail of (at zero or infinity) leads to an interestingcomparison of M_t with the largest jump of X_t in (0, t]. Some results concerning the convergence at zero and infinityof integrals like t g(a(t) + |X_t|) dt, t g(S_t) dt,and t g(R_t) dt, where S_t is the supremum process and R_t= S_t – X_t is the process reflected in its supremum, arealso given. We also consider the convergence of integrals suchas , etc.     

A Functional Relation for Accessory Parameters for Genus 2 Algebraic Curves with an Order 4 Automorphism

Let C be a genus 2 algebraic curve defined by an equation ofthe form y² = x(x² – 1)(x – a)(x – 1/a). Asis well known, the five accessory parameters for such an equationcan all be expressed in terms of a and the accessory parameter b corresponding to a. The main result of the paper is thatif a' = 1 – a², which in general yields a non-isomorphiccurve C', then b'a'(a'² – 1) = – – ba(a²– 1). This is proven by it being shown how the uniformizing functionfrom the unit disk to C' can be explicitly described in termsof the uniformizing function for C.     

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.     

Stability in the Sense of Bounded Average Power

An R^m-valued sequence (x_k): = (x_k : k = 1, 2, ...), e.g. generatedrecursively by x_k = f_k (x_k–_k, U_k), is called ‘averagepth power bounded’ if (1/K) is bounded uniformly in K= 1, 2,.... (The case p = 2 may correspond to ‘power’in the physical sense.) This is a notion of stability. Givenestimates of the form: f_k (x, u) < a x + ¶ _k conditionsare obtained on the coefficient sequence (a_k) and the inputestimates e_k:=¶_k (u_k) which ensure this form of stabilityfor the output (x_k). In particular, a condition (utilized inan application to adaptive control) is obtained which imposes(i) a bound b on (a_k) and a ‘sparsity measure’ m(K) on #{kK: a_k>} as K ( >1) (ii) average pth power boundednesson (e_k), and (iii) a growth condition on (e_k) related to b andm (•). This condition is sharp.     

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.     

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.     

The Heat Flow of the CCR Algebra

Let Pf(x) = –if'(x) and Qf(x) = xf(x) be the canonicaloperators acting on an appropriate common dense domain in L²(R).The derivations D_P(A) = i(PA–AP) and D_Q(A) = i(QA–AQ)act on the *-algebra A of all integral operators having smoothkernels of compact support, for example, and one may considerthe noncommutative ‘Laplacian’, L = + , as a linear mapping of A into itself. L generates a semigroup of normal completely positive linearmaps on B(L₂(R)), and this paper establishes some basic propertiesof this semigroup and its minimal dilation to an E₀-semigroup.In particular, the author shows that its minimal dilation ispure and has no normal invariant states, and he discusses thesignificance of those facts for the interaction theory introducedin a previous paper. There are similar results for the canonical commutation relationswith n degrees of freedom, where 1 n < . 2000 MathematicsSubject Classification 46L57 (primary), 46L53, 46L65 (secondary).     

On Lp boundedness of wave operators for 4-dimensional Schrodinger operators with threshold singularities

Let H=–+V(x) be a Schrödinger operator on L²(R⁴),H₀=–. Assume that |V(x)|+| V(x)|C x ^– for some>8. Let be the wave operators. It is known that W_± extend to bounded operators in L^p(R⁴)for all 1p, if 0 is neither an eigenvalue nor a resonance ofH. We show that if 0 is an eigenvalue, but not a resonance ofH, then the W_± are still bounded in L^p(R⁴) for all psuch that 4/3<p<4.     

The density of prime divisors in the arithmetic dynamics of quadratic polynomials

Let f [x], and consider the recurrence given by a_n = f(a_{n –1}), with a₀ . Denote by P(f, a₀) the set of prime divisorsof this recurrence, that is, the set of primes dividing at leastone non-zero term, and denote the natural density of this setby D(P(f, a₀)). The problem of determining D(P(f, a₀)) whenf is linear has attracted significant study, although it remainsunresolved in full generality. In this paper, we consider thecase of f quadratic, where previously D(P(f, a₀)) was knownonly in a few cases. We show that D(P(f, a₀)) = 0 regardlessof a₀ for four infinite families of f, including f = x² + k,k \{–1}. The proof relies on tools from group theoryand probability theory to formulate a sufficient condition forD(P(f, a₀)) = 0 in terms of arithmetic properties of the forwardorbit of the critical point of f. This provides an analogy toresults in real and complex dynamics, where analytic propertiesof the forward orbit of the critical point have been shown todetermine many global dynamical properties of a quadratic polynomial.The article also includes apparently new work on the irreducibilityof iterates of quadratic polynomials.     

Necessary and Sufficient Conditions for Exponential Stability and Ultimate Boundedness of Systems Governed by Stochastic Partial Differential Equations

Consider the following infinite dimensional stochastic evolutionequation over some Hilbert space H with norm |·|: It is proved that under certain mild assumptions, the strongsolution X_t(x₀)VHV*, t 0, is mean square exponentially stableif and only if there exists a Lyapunov functional (·,·):HxR₊R¹ which satisfies the following conditions: (i)c₁|x|²–k₁e^–µ₁t(x,t)c₂|x|²+k₂+k₂e^–µ₂t; (ii) L(x,t)–c₃(x,t)+k₃e^–µ₃t, xV, t0; where L is the infinitesimal generator of the Markov processX_t and c_i, k_i, µ_i, i = 1, 2, 3, are positive constants.As a by-product, the characterization of exponential ultimateboundedness of the strong solution is established as the nulldecay rates (that is, µ_i = 0) are considered.     

Some Integrals of Gamma, Polygamma and Volterra Functions

Infinite integrals of the gamma and polygamma functions areexpressed in terms of the Volterra and related functions usingthe Laplace transformation method. A number of infinite, convolution,indefinite and definite integrals of v(x), v(x, ), µ(x,ß) and µ(x, ß, ) has been evaluated.A short table of the Volterra function v(x) in the range 0 x 10–0 is presented.     

On a model of viscoelastic rod in unilateral contact with a rigid wall

^** Corresponding author. Email: atanackovic{at}uns.ns.ac.yu We study translatory motion of a body to which a viscoelasticrod with the constitutive equation with fractional derivativesis attached. The body with a rod impacts against a rigid wall.It is shown that the problem is described with a coupled systemof differential equations having integer and fractional derivativeshaving the form x⁽²⁾ = –f; f + af⁽⁾ = x + bx⁽⁾, x(0) =0, x⁽¹⁾(0) = 1. The unique solvability in S'₊ is proved andinterpretation of solutions is given. Also, some a priori estimatesof the solution are given. In particular, we showed that restrictionson coefficients that follow from the second law of thermodynamicsimply that the velocity after the impact is smaller than thevelocity before the impact.     

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.     

Concavity and B-Concavity of Solutions of Quasilinear Filtration Equations

Spatial concavity properties of non-negative weak solutionsof the filtration equations with absorption u_t = ((u))_xx–(u)in Q = Rx(0, ), '0, 0 are studied. Under certain assumptionson the coefficients , it is proved that concavity of the pressurefunction is a consequence of a ‘weak’ convexityof travelling-wave solutions of the form V(x, t) = (x–t+a).It is established that the global structure of a so-called properset B = {V} of such particular solutions determines a propertyof B-concavity for more general solutions which is preservedin time. For the filtration equation u_t = ((u))_xx a semiconcavityestimate for the pressure, v_xx(t+)^–1'(), due to the B-concavityof the solution to the subset B of the explicit self-similarsolutions (x/t+)) is proved. The analysis is based on the intersection comparison based onthe Sturmian argument of the general solution u(x, t) with subsetsB of particular solutions. Also studied are other aspects ofthe B-concavity/convexity with respect to different subsetsof explicit solutions.