Generalized Logarithmic Derivative Estimates of Gol'dberg-Grinshtein Type

For f meromorphic in the complex plane and meromorphic in theunit disc, sharp upper bounds are obtained for and where k andj are integers satisfying k > j 0. The results generalizethe logarithmic derivative estimate due to Gol'dberg and Grinshteinto derivatives of higher order. 2000 Mathematics Subject Classification30D35.     

Economical Covers with Geometric Applications

A cover of a hypergraph is a collection of edges whose unioncontains all vertices. Let H = (V, E) be a k-uniform, D-regularhypergraph on n vertices, in which no two vertices are containedin more than o(D/e^2k log D) edges as D tends to infinity. Ourresults include the fact that if k = o(log D), then there isa cover of (1 + o(1))n/k edges, extending the known result thatthis holds for fixed k. On the other hand, if k 4 log D thenthere are k-uniform, D-regular hypergraphs on n vertices inwhich no two vertices are contained in more than one edge, andyet the smallest cover has at least (nk) log (k log D)) edges.Several extensions and variants are also obtained, as well asthe following geometric application. The minimum number of linesrequired to separate n random points in the unit square is,almost surely, (n^2/3 / (log n)^1/3). 2000 Mathematical SubjectClassification: 05C65, 05D15, 60D05.     

Exceptional Functions and Normality

Yang proved in [10] that if f and f^(k) have no fix-points forevery fF, where F is a family of meromorphic functions in adomain G and k a fixed integer, then F is normal in G. In thispaper we prove normality for families F for which every fF omits₁ and f^(k) omits ₂, where ₁ and ₂ are analytic functions with. 1991 Mathematics SubjectClassification 30D35, 30D45.     

Normal families and omitted functions II

Let k 2 be an integer and let be a family of functions meromorphicon a domain D in , all of whose poles are multiple and whosezeros all have multiplicity at least k + 1. Let h be a functionmeromorphic on D, h 0, . Suppose that for each f , f^(k)(z) h(z) for z D. Then is a normal family on D.     

Application of Spline Functions to Systems of Volterra Integral Equations of the First and Second Kinds

Continuous approximations to the solution of systems of Volterraintegral equations of the first and second kinds are soughtby methods using spline functions of degree m, deficiency-(k—1),i.e. in C^m—k, and a fixed quadrature rule of degree p-1,p m-1. The resulting method is called an (m, k)-method. Thestability behaviour of the (m, 1)- and the (m, m)-method isstudied for arbitrarily finite m. Also studied is the stabilityof the (m, m-1)-method for second-kind systems. Convergenceresults and asymptotic formulae for the discretization errorare obtained.     

Rational Points on a Class of Superelliptic Curves

A famous Diophantine equation is given by y^k=(x+1)(x+2)...(x+m). (1) For integers k2 and m2, this equation only has the solutionsx = –j (j = 1, ..., m), y = 0 by a remarkable result ofErds and Selfridge [9] in 1975. This put an end to the old questionof whether the product of consecutive positive integers couldever be a perfect power (except for the obviously trivial cases).In a letter to D. Bernoulli in 1724, Goldbach (see [7, p. 679])showed that (1) has no solution with x0 in the case k = 2 andm = 3. In 1857, Liouville [18] derived from Bertrand's postulatethat for general k2 and m2, there is no solution with x0 ifone of the factors on the right-hand side of (1) is prime. Byuse of the Thue–Siegel theorem, Erds and Siegel [10] provedin 1940 that (1) has only trivial solutions for all sufficientlylarge kk₀ and all m. This was closely related to Siegel's earlierresult [30] from 1929 that the superelliptic equation y^k=f(x) has at most finitely many integer solutions x, y under appropriateconditions on the polynomial f(x). The ineffectiveness of k₀was overcome by Baker's method [1] in 1969 (see also [2]). In 1955, Erds [8] managed to re-prove the result jointly obtainedwith Siegel by elementary methods. A refinement of Erds' ideasfinally led to the above-mentioned theorem as follows.     

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.     

Strongly n-Trivial Knots

The authors prove in this paper that, given any knot k of genusg,k fails to be strongly n-trivial for all n with n 6g –3. 2000 Mathematics Subject Classification 57M99.     

Repulsive Fixpoints of Analytic Functions with Applications to Complex Dynamics

Let G be a family of functions analytic in a domain D in thecomplex plane. It is proved that G is a normal family, providedthat for each fG, there exists k = k(f) > 1 such that thekth iterate f^k has no repulsive fixpoint in D. A new proof ofa result of Bergweiler and Terglane concerning the dynamicsof entire functions is also given.     

Conditions for the Solvability of Systems of Two and Three Additive Forms Over p-Adic Fields

This paper is concerned with non-trivial solvability in p-adicintegers of systems of two and three additive forms. Assumingthat the congruence equation ax^k + by^k + cz^k d (modp) has asolution with xyz 0(modp) we have proved that any system oftwo additive forms of odd degree k with at least 6k + 1 variables,and any system of three additive forms of odd degree k withat least 14k + 1 variables always has non-trivial p-adic solutions,provided p does not divide k. The assumption of the solubilityof the congruence equation above is guaranteed for example ifp > k⁴. In the particular case of degree k = 5 we have proved the followingresults. Any system of two additive forms with at least n variablesalways has non-trivial p-adic solutions provided n 31 and p> 101 or n 36 and p > 11. Furthermore any system of threeadditive forms with at least n variables always has non-trivialp-adic solutions provided n 61 and p > 101 or n 71 andp > 11. 2000 Mathematics Subject Classification 11D72, 11D79.     

The Hausdorff Dimension of Julia Sets of Meromorphic Functions II

A family of transcendental meromorphic functions, f_p(z), p N is considered. It is shown that, if p 6, then the Hausdorffdimension of the Julia set of f_p satisfies dim J(f_p) 1/p, for0 < < 1/6^p, and dim J(f_p) 1–(30 ln ln p/ln p),for p^4p–1/10⁵ ln p < < p^4p–1/10⁴ ln p. Theseresults are used elsewhere to show that, for each d (0, 1),there exists a transcendental meromorphic function for whichdim J(f) = d.     

Shared values and normality

This paper investigates the relationship between the normality and the shared values for a meromorphic function on the unit disc Δ. Based on Marty’s normality criterion and through a detailed analysis of the meromorphic functions, it is shown that if for every f ∈ , f and f ^(k) share a and b on Δ and the zeros of f(z) − a are of multiplicity k ⩾ 3, then is normal on Δ, where is a family of meromorphic functions on the unit disc Δ, and a and b are distinct values. Selected from Journal of East China Normal University (Natural Science), 2003, 4: 12–18. This work was supported by the National Natural Science Foundation of China under grant number 10271122 and by Shanghai City Foundation for selected academic research     

5-Torsion Points on Curves of Genus 2

Let C be a smooth proper curve of genus 2 over an algebraicallyclosed field k. Fix a Weierstrass point in C(k) and identifyC with its image in its Jacobian J under the Albanese embeddingthat uses as base point. For any integer N1, we write J_N forthe group of points in J(k) of order dividing N and for the subset of J_N of points oforder N. It follows from the Riemann–Roch theorem thatC(k)J₂ consists of the Weierstrass points of C and that C(k) and C(k) are empty (see [3]). The purpose of this paper is to study curvesC with C(k) non-empty.     

Convolution complementarity problems with application to impact problems

^** Email: dstewart{at}math.uiowa.edu. Part of this work was carried out while visiting CMAF at the University of Lisbon and while visiting the University of Lyons 1. Convolution complementarity problems (CCPs) have the followingform: given a matrix-valued function k and a vector-valued functionq, find a vector-valued function u satisfying 0 u(t) (k*u)(t)+ q(t) 0 for all t. In this paper CCPs are applied to a mechanicalimpact problem, but they can also be applied to other dynamicproblems with hard constraints. CCPs are shown to have solutionsprovided q(0) 0 and q is sufficiently regular, k has locallybounded variation and k(0⁺) is a P-matrix. Uniqueness also holdsprovided, in addition, k(0⁺) is symmetric positive definite.This theory shows that the impact problem studied here has aunique solution, and that energy is conserved. Numerical methodshave been devised and implemented for the impact problem, andthe results are presented.     

Annular Functions and Gap Series

If f(z) = c_kz^nk, where n_k+1/n_k q > 1, and f(z) is analyticin |z| < 1, the f(z) is an annular function if and only ifsup |c_k| = . This answers a question posed by L. R. Sons andD.M. Campbell simplifies the proofs of many known examples ofannular functions. Present address: Dept. of Mathematical Sciences, McMaster University,Hamilton, Ontario, Canada L8S4K1     

Determination of a Convex Body from Minkowski Sums of its Projections

For a convex body K in R^d and 1 K d – 1, let P_K (K)be the Minkowski sum (average) of all orthogonal projectionsof K onto k-dimensional subspaces of R^d. It is Known that theoperator P_k is injective if kd/2, k=3 for all d, and if k =2, d 14. It is shown that P_2k (K) determines a convex body K among allcentrally symmetric convex bodies and P_2k+1(K) determines aconvex body K among all bodies of constant width. Correspondingstability results are also given. Furthermore, it is shown thatany convex body K is determined by the two sets P_k (K) and P_k'(K) if 1 < k < k'. Concerning the range of P_k , 1 k d–2, it is shown that its closure (in the Hausdorff-metric)does not contain any polytopes other than singletons.     

A Graph-Theoretic Approach to the Unique Midset Property of Metric Spaces

A metric space X has the unique midset property if there isa topology-preserving metric d on X such that for every pairof distinct points x, y there is one and only one point p suchthat d(x, p) = d(y, p). The following are proved. (1) The discretespace with cardinality n has the unique midset property if andonly if n 2, 4 and n c, where c is the cardinality of thecontinuum. (2) If D is a discrete space with cardinality notgreater than c, then the countable power D^N of D has the uniquemidset property. In particular, the Cantor set and the spaceof irrational numbers have the unique midset property. A finite discrete space with n points has the unique midsetproperty if and only if there is an edge colouring of the completegraph K_n such that for every pair of distinct vertices x, ythere is one and only one vertex p such that (xp) = (yp). Letump(K_n) be the smallest number of colours necessary for sucha colouring of K_n. The following are proved. (3) For each k 0, ump(K_2k+1) = k. (4) For each k 3, k ump(K_2k) 2k–1.     

On the extension of real regular embedding

Let X be a real nonsingular affine algebraic variety of dimensionk. It is proved that any two regular (algebraic) embeddingsX ⁿ are regularly equivalent, provided that n 4k + 2.     

(k, l)-Algebraic Stability of Runge-Kutta Methods

For ordinary differential equations satisfying a one-sided Lipschitzcondition with Lipschitz constant v, the solutions satisfy with l=hv, so that, in the case of Runge-Kutta methods, estimatesof the form ||y_n||²k(l)||y_n–1||² are desirable. Burrage(1986) has investigated the behaviour of the error-boundingfunction k for positive l for the family of s-stage Gauss methodsof order 2s, and has shown that k(l)=exp 2l+O(l³) (l0) for s3.In this paper, we extend the analysis of k to any irreduciblealgebraically stable Runge-Kutta method, and obtain resultsabout the maximum order of k as an approximation to exp 2l.As a particular example, we investigate the function k for allalgebraically stable methods of order 2s–1.     

Asymptotics for the Heat Content of a Planar Region with a Fractal Polygonal Boundary

Let k 3 be an integer. For 0<s<1, let D_s R² be the setthat is constructed iteratively as follows. Take a regular openk-gon with sides of unit length, attach regular open k-gonswith sides of length s to the middles of the edges, and so on.At each stage of the iteration the k-gons that are added area factor s smaller than the previous generation and are attachedto the outer edges of the family grown so far. The set D_s isdefined to be the interior of the closure of the union of allthe k-gons. It is easy to see that there must exist some s_k> 0 such that no k-gons overlap if and only if 0 < s s_k. We derive an explicit formula for s_k. The set D_s is open, bounded, connected and has a fractal polygonalboundary. Let denote the heat content of D_s at time t when D_s initially has temperature 0and D_s is kept at temperature 1. We derive the complete short-timeexpansion of up to terms that are exponentially small in 1/t. It turns out that there arethree regimes, corresponding to 0<s<1/(k–1), s=1/(k–1),and 1/(k–1)<s s_k. For s 1/(k–1) the expansionhas the form where p_s is a log (1/s²)-periodic function, d_s=log (k–1)/log(1/s) is a similarity dimension, A_s and B are constants relatedto the edges and vertices, respectively, of D_s, and r_s is anerror exponent. For s=1/(k–1), the t^1/2-term carries anadditional log t. 1991 Mathematics Subject Classification: 11D25,11G05, 14G05.