On diagonal cubic surfaces

We show that the affine surfacesx ³+y ³+c z ³=c, c Q, in the casesc2,c=2, contain precisely 2, respectively 4, polynomial parametric solutions corresponding to curves of arithmetic genus 0 on the surface.However, these surfaces contain infinitely many polynomial parametric solutions corresponding to curves of arithmetic genus greater than 0.The author wishes to acknowledge the receipt of a Summer Support Grant for the College of Liberal Arts, Arizona State University, while this note was being written.     

Maximality properties for isometric interpolating sequences and sequences of trivial points in the spectrum of H∞

Let (x_n) be an isometric interpolating sequence or a sequence of trivial points in the spectrum of H^∞. It is shown that either every cluster point of that sequence has a maximal support set or there exists y ∈ M(H^∞+C) such that the support of x_n is contained in the support of y for infinitely many n. Similar results for Gleason parts are obtained, too. We also investigate the H^∞‐convex hulls of countable unions of support sets and show that whenever supp x ? supp y and x /∈ , then the H^∞‐convex hull of supp x does not meet . (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Counterexamples to faudree and schelp's conjecture on hamiltonian-connected graphs

Faudree and Schelp conjectured that for any two vertices x, y in a Hamiltonian-connected graph G and for any integer k, where n/2 ? k ? n ? 1, G has a path of length k connecting x and y. However, we show in this paper that there are infinitely many exceptions to this conjecture and we comment on some problems on path length distribution raised by Faudree and Schelp.     

Equations fonctionnelles et estimations de normes de matrices

We solve independently the equations 1/θ(x)θ(y)=ψ(x)−ψ(y)+φ(x−y)/θ(x−y) and 1/θ(x)θ(y)=σ(x)−σ(y)/θ(x−y)+τ(x)τ(y), τ(0)=0. In both cases we find θ²=aθ⁴+bθ²+c. We deduce estimates for the spectral radius of a matrix of type(1/θ(x _r −x _s )) (the accent meaning that the coefficients of the main diagonal are zero) and we study the case where thex _r are equidistant.

Dédié to à Monsieur le Professeur Otto Haupt à l'occasion de son cententiare avec les meilleurs voeux     

On arithmetic progressions on Pellian equations

We consider arithmetic progressions consisting of integers which are y-components of solutions of an equation of the form x ² ? dy ² = m. We show that for almost all four-term arithmetic progressions such an equation exists. We construct a seven-term arithmetic progression with the given property, and also several five-term arithmetic progressions which satisfy two different equations of the given form. These results are obtained by studying the properties of a parametric family of elliptic curves.     

On rational approximations with denominators from thin sets, ii

In the general case a real irrational number cannot be approximated by infinitely many rationalsp/q involving error terms less than q^-2 when the denominatorsq are taken from a given thin set of positive integers. The distribution of irrationals which are situated in close neighborhoods of infinitely many fractionsp/q, whereq is restricted to the elements of a thin set, depends on the asymptotic behaviour of theq’s and on their arithmetic properties.     

Approximation properties of zonal function networks using scattered data on the sphere

A zonal function (ZF) network is a function of the form x↦∑ _k=1 ⁿ c _k (x · y _k), where x and the y _k's are on the unit sphere in q+1 dimensional Euclidean space, and where the y _k's are scattered points. In this paper, we study the degree of approximation by ZF networks. In particular, we compare this degree of approximation with that obtained with the classical spherical harmonics. In many cases of interest, this is the best possible for a given amount of information regarding the target function. We also discuss the construction of ZF networks using scattered data. Our networks require no training in the traditional sense, and provide theoretically predictable rates of approximation. This revised version was published online in June 2006 with corrections to the Cover Date.     

Unbounded Solutions of Almost Periodically Forced Pendulum-Type Equations

In this paper, we prove that the forced pendulum-type equation , where with 1-periodicity in x satisfies the conditions: and , possesses infinitely many unbounded solutions on a cylinder S ¹×R for any almost periodic function p(t) with nonvanishing mean value. Received October 7, 1997, Accepted June 9, 1999     

Semi-empirical likelihood confidence intervals for the differences of quantiles with missing data

Detecting population （group） differences is useful in many applications, such as medical research. In this paper, we explore the probabilistic theory for identifying the quantile differences .between two populations. Suppose that there are two populations x and y with missing data on both of them, where x is nonparametric and y is parametric. We are interested in constructing confidence intervals on the quantile differences of x and y. Random hot deck imputation is used to fill in missing data. Semi-empirical likelihood confidence intervals on the differences are constructed.     

On Diophantine equation 3a 2 x 4 − By 2 = 1

Bumby proved that the only positive solutions to the quartic Diophantine equation 3x ⁴ ? 2y ² = 1 are (x, y) = (1, 1), (3, 11). In this paper, we extend this result and prove that if the class number of the field ${{\rm Q}(\sqrt{1-3a^{2}})}Bumby proved that the only positive solutions to the quartic Diophantine equation 3x ⁴ − 2y ² = 1 are (x, y) = (1, 1), (3, 11). In this paper, we extend this result and prove that if the class number of the field Q(?{1-3a²}){{\rm Q}(\sqrt{1-3a^{2}})} is not divisible by 2, the equation 3a ² x ⁴ − By ² = 1 has at most two solutions. However, both solutions occur in only one case, a = 1, b = 2, as solved by Bumby. The proof utilizes the law of quadratic reciprocity that seems very rare in solving Diophantine equations, and the solution will be also obtained effectively through the proof when it exists.     

On –δu = K(x)u5 in ℝ3

In this paper, we study the equation –Δu = K(x)u⁵ in ?³ and provide a large class of positive functions K(x) for which we obtain infinitely many positive solutions which decay at infinity at the rate of |x|^?1. © 1993 John Wiley & Sons, Inc.     

Well/Ill Posedness for the Euler-Korteweg-Poisson System and Related Problems

We consider a general Euler-Korteweg-Poisson system in R ³, supplemented with the space periodic boundary conditions, where the quantum hydrodynamics equations and the classical fluid dynamics equations with capillarity are recovered as particular examples. We show that the system admits infinitely many global-in-time weak solutions for any sufficiently smooth initial data including the case of a vanishing initial density - the vacuum zones. Moreover, there is a vast family of initial data, for which the Cauchy problem possesses infinitely many dissipative weak solutions, i.e. the weak solutions satisfying the energy inequality. Finally, we establish the weak-strong uniqueness property in a class of solutions without vacuum. In this paper we show that, even in presence of a dispersive tensor, we have the same phenomena found by De Lellis and Székelyhidi.     

Taking pth Roots Modulo Polynomials over Finite Fields

In this contribution we show how to find y(x) in the polynomial equation y(x) ^p t(x) mod f(x), where t(x), y(x) and f(x) are polynomials over the field GF(p ^m). The solution of such equations are thought for in many cases, e.g., for p = 2 it is a step in the so-called Patterson Algorithm for decoding binary Goppa codes.     

Computing the Selmer group of an isogeny between Abelian varieties using a further isogeny to a Jacobian

This article describes an algorithm for computing the Selmer group of an isogeny between abelian varieties. This algorithm applies when there is an isogeny from the image abelian variety to the Jacobian of a curve. The use of an auxiliary Jacobian simplifies the determination of locally trivial cohomology classes. An example is presented where the rational solutions to x⁴+(y²+1)(x+y)=0 are determined.     

A nilpotent group and its elliptic curve: Non-uniformity of local zeta functions of groups

A nilpotent group is defined whose local zeta functions counting subgroups and normal subgroups depend on counting points modp on the elliptic curvey ²=x ³−x. This example answers negatively a question raised in the paper of F. J. Grunewald, D. Segal and G. C. Smith where these local zeta functions were first defined. They speculated that local zeta functions of nilpotent groups might be finitely uniform asp varies. A proof is given that counting points on the elliptic curvey ²=x ³−x are not finitely uniform, and hence the same is true for the zeta function of the associated nilpotent group. This example demonstrates that nilpotent groups have a rich arithmetic beyond the connection with quadratic forms.     

On linear forms whose coefficients are in arithmetic progression

Let Ω be the set of positive integers that are omitted values of the form where thea _i are fixed positive integers with g.c.d. 1 and thex _i are variable nonnegative integers. Set ω=|Ω| and κ=max Ω+1. Using an expression of Roberts [4] for κ when thea _i form an arithmetic progression, we determine ω in this case.     

A note on the congruences of the nakano superlattice and some properties of the associated quotients

In this paper we explore theNakano superlattice (H, ⊔, ⊓), where ⊔, ⊓ are the Nakanohyperoperations x⊔y={z:x∨z=y∨z=x∨y},x⊓y={z:x∧z=y∧z=x∧y}. In particular, we study the properties of congruences on the Nakano superlattice and the associated quotients. New hyperoperations are introduced on the quotient and their properties studied.     

The commutation relation <Emphasis Type="Italic">xy</Emphasis>=<Emphasis Type="Italic">qyx</Emphasis>+<Emphasis Type="Italic">hf</Emphasis>(<Emphasis Type="Italic">y</Emphasis>) and Newton’s binomial formula

Let x and y be two variables satisfying the commutation relation xy=qyx+hf(y), where f(y) is a polynomial. In this paper, using Young diagrams and generating functions techniques, we study the binomial formula (x+y) ⁿ and we present an identity for x ^m y. The connection to Operator Calculus is discussed and several special cases are treated explicitly.     

Global Ill‐Posedness of the Isentropic System of Gas Dynamics

We consider the isentropic compressible Euler system in 2 space dimensions with pressure law p (ρ) = ρ² and we show the existence of classical Riemann data, i.e. pure jump discontinuities across a line, for which there are infinitely many admissible bounded weak solutions (bounded away from the void). We also show that some of these Riemann data are generated by a 1‐dimensional compression wave: our theorem leads therefore to Lipschitz initial data for which there are infinitely many global bounded admissible weak solutions. © 2015 Wiley Periodicals, Inc.     

Interval orders and circle orders

A finite poset is an interval order if its point can be mapped into real intervals so that x in the poset precisely when x's interval lies wholly to the left of y's; the poset is a circle order if its points can be mapped into circular disks in the plane so that x precisely when x's circular disk is properly included in y's. This note proves that every finite interval order is a circle order.