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Volume 85 - June 2012

Contents

Page 2 of 3

Research Article

  • BIHARMONIC SUBMANIFOLDS IN NONFLAT LORENTZ 3-SPACE FORMS

  • Part of:
  • TORU SASAHARA
  • Published online by Cambridge University Press:
    30 November 2011, pp. 422-432
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  • The purpose of this paper is to classify nonharmonic biharmonic curves and surfaces in de Sitter 3-space and anti-de Sitter 3-space.

  • SUMSETS AND DIFFERENCE SETS CONTAINING A COMMON TERM OF A SEQUENCE

  • Part of:
  • QUAN-HUI YANG, YONG-GAO CHEN
  • Published online by Cambridge University Press:
    26 September 2011, pp. 79-83
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  • Let β>1 be a real number, and let {ak} be an unbounded sequence of positive integers such that ak+1/ak≤β for all k≥1. The following result is proved: if n is an integer with n>(1+1/(2β))a1 and A is a subset of {0,1,…,n} with , then (A+A)∩(A−A) contains a term of {ak }. The lower bound for |A| is optimal. Beyond these, we also prove that if n≥3 is an integer and A is a subset of {0,1,…,n} with , then (A+A)∩(A−A) contains a power of 2. Furthermore, cannot be improved.

  • ON BAD SUPERNILPOTENT RADICALS

  • Part of:
  • H. FRANCE-JACKSON
  • Published online by Cambridge University Press:
    15 December 2011, pp. 271-274
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  • A supernilpotent radical α is called bad if the class π(α) of all prime and α-semisimple rings consists of the one-element ring 0 only. We construct infinitely many bad supernilpotent radicals which form a generalization of Ryabukhin’s example of a supernilpotent nonspecial radical. We show that the family of all bad supernilpotent radicals is a sublattice of the lattice of all supernilpotent radicals and give examples of supernilpotent radicals that are not bad.

  • AN ARBITRARY INTERSECTION OF Lp-SPACES

  • Part of:
  • F. ABTAHI, H. G. AMINI, H. A. LOTFI, A. REJALI
  • Published online by Cambridge University Press:
    16 February 2012, pp. 433-445
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  • For a locally compact group G and an arbitrary subset J of [1,∞], we introduce ILJ(G) as a subspace of ⋂ p∈JLp(G) with some norm to make it a Banach space. Then, for some special choice of J, we investigate some topological and algebraic properties of ILJ(G) as a Banach algebra under a convolution product.

  • ASYMPTOTIC RESULTS FOR THE NUMBER OF PATHS IN A GRID

  • Part of:
  • ALOIS PANHOLZER, HELMUT PRODINGER
  • Published online by Cambridge University Press:
    04 October 2011, pp. 446-455
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  • In two recent papers, Albrecht and White [‘Counting paths in a grid’, Austral. Math. Soc. Gaz.35 (2008), 43–48] and Hirschhorn [‘Comment on “Counting paths in a grid”’, Austral. Math. Soc. Gaz.36 (2009), 50–52] considered the problem of counting the total number Pm,n of certain restricted lattice paths in an m×n grid of cells, which appeared in the context of counting train paths through a rail network. Here we give a precise study of the asymptotic behaviour of these numbers for the square grid, extending the results of Hirschhorn, and furthermore provide an asymptotic equivalent of these numbers for a rectangular grid with a constant proportion α=m/n between the side lengths.

  • THE BOUNDARY VOLUME OF A LATTICE POLYTOPE

  • Part of:
  • GÁBOR HEGEDÜS, ALEXANDER M. KASPRZYK
  • Published online by Cambridge University Press:
    26 September 2011, pp. 84-104
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  • For a d-dimensional convex lattice polytope P, a formula for the boundary volume vol(∂P) is derived in terms of the number of boundary lattice points on the first ⌊d/2⌋ dilations of P. As an application we give a necessary and sufficient condition for a polytope to be reflexive, and derive formulas for the f-vector of a smooth polytope in dimensions three, four, and five. We also give applications to reflexive order polytopes, and to the Birkhoff polytope.

  • ON THE ALGEBRAIC CONVERGENCE OF FINITELY GENERATED KLEINIAN GROUPS IN ALL DIMENSIONS

  • Part of:
  • XI FU
  • Published online by Cambridge University Press:
    08 December 2011, pp. 275-279
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  • Let {Gr,i} be a sequence of r-generator Kleinian groups acting on . In this paper, we prove that if {Gr,i} satisfies the F-condition, then its algebraic limit group Gr is also a Kleinian group. The existence of a homomorphism from Gr to Gr,i is also proved. These are generalisations of all known corresponding results.

  • A VARIATIONAL McSHANE INTEGRAL CHARACTERISATION OF THE WEAK RADON–NIKODYM PROPERTY

  • Part of:
  • SOKOL BUSH KALIAJ
  • Published online by Cambridge University Press:
    21 February 2012, pp. 456-462
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  • We present a characterisation of Banach spaces possessing the weak Radon–Nikodym property in terms of finitely additive interval functions whose McShane variational measures are absolutely continuous with respect to Lebesgue measure.

  • RATIONAL POINTS ON THREE SUPERELLIPTIC CURVES

  • Part of:
  • ZHONGYAN SHEN, TIANXIN CAI
  • Published online by Cambridge University Press:
    06 September 2011, pp. 105-113
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  • In this paper, we obtain all rational points (x,y) on the superelliptic curves

  • UNIQUENESS OF MEROMORPHIC FUNCTIONS SHARING ONE VALUE

  • Part of:
  • CHUN WU, CHUNLAI MU, JIANGTAO LI
  • Published online by Cambridge University Press:
    04 January 2012, pp. 280-294
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  • In this paper, we study the uniqueness of meromorphic functions concerning differential polynomials sharing nonzero finite values, and obtain some results which improve the results of Yang and Hua, Xu and Qiu, Fang and Hong, and Dyavanal, among others.

  • NOTE ABOUT LINDELÖF Σ-SPACES υX

  • Part of:
  • J. KA̧KOL, M. LÓPEZ-PELLICER
  • Published online by Cambridge University Press:
    28 September 2011, pp. 114-120
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  • The paper deals with the following problem: characterize Tichonov spaces X whose realcompactification υX is a Lindelöf Σ-space. There are many situations (both in topology and functional analysis) where Lindelöf Σ (even K-analytic) spaces υX appear. For example, if E is a locally convex space in the class 𝔊 in sense of Cascales and Orihuela (𝔊 includes among others (LM ) -spaces and (DF ) -spaces), then υ(E′,σ(E′,E)) is K-analytic and E is web-bounded. This provides a general fact (due to Cascales–Kakol–Saxon): if E∈𝔊, then σ(E′,E) is K-analytic if and only if σ(E′,E) is Lindelöf. We prove a corresponding result for spaces Cp (X) of continuous real-valued maps on X endowed with the pointwise topology: υX is a Lindelöf Σ-space if and only if X is strongly web-bounding if and only if Cp (X) is web-bounded. Hence the weak* dual of Cp (X) is a Lindelöf Σ-space if and only if Cp (X) is web-bounded and has countable tightness. Applications are provided. For example, every E∈𝔊 is covered by a family {Aα :α∈Ω} of bounded sets for some nonempty set Ω⊂ℕℕ.

  • ON PROPERTIES OF FINITE-ORDER MEROMORPHIC SOLUTIONS OF A CERTAIN DIFFERENCE PAINLEVÉ I EQUATION

  • Part of:
  • MEI-RU CHEN, ZONG-XUAN CHEN
  • Published online by Cambridge University Press:
    08 December 2011, pp. 463-475
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  • In this paper, we investigate properties of finite-order transcendental meromorphic solutions, rational solutions and polynomial solutions of the difference Painlevé I equation where a, b and c are constants, ∣a∣+∣b∣≠0.

  • UNIQUENESS OF ENTIRE FUNCTIONS SHARING A VALUE WITH LINEAR DIFFERENTIAL POLYNOMIALS

  • Part of:
  • INDRAJIT LAHIRI, RAJIB MUKHERJEE
  • Published online by Cambridge University Press:
    30 November 2011, pp. 295-306
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  • We study the uniqueness of entire functions sharing a nonzero finite value with linear differential polynomials and improve a result of P. Li.

  • ESSENTIAL NORM OF EXTENDED CESÀRO OPERATORS FROM ONE BERGMAN SPACE TO ANOTHER

  • Part of:
  • ZHANGJIAN HU
  • Published online by Cambridge University Press:
    12 December 2011, pp. 307-314
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  • Let Ap(φ) be the pth Bergman space consisting of all holomorphic functions f on the unit ball B of ℂn for which , where φ is a given normal weight. Let Tg be the extended Cesàro operator with holomorphic symbol g. The essential norm of Tg as an operator from Ap (φ) to Aq (φ) is denoted by . In this paper it is proved that, for p≤q, with 1/k=(1/p)−(1/q) , where ℜg(z) is the radial derivative of g; and for p>q, with 1/s=(1/q)−(1/p) .

  • ITÔ’S THEOREM ON GROUPS WITH TWO CLASS SIZES REVISITED

  • Part of:
  • ELENA ALEMANY, ANTONIO BELTRÁN, MARÍA JOSÉ FELIPE
  • Published online by Cambridge University Press:
    19 October 2011, pp. 476-481
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  • Let G be a finite p-solvable group. We prove that if G has exactly two conjugacy class sizes of p′-elements of prime power order, say 1 and m, then m=paqb, for two distinct primes p and q, and G either has an abelian p-complement or G=PQ×A, with P and Q a Sylow p-subgroup and a Sylow q-subgroup of G, respectively, and A is abelian. In particular, we provide a new extension of Itô’s theorem on groups having exactly two class sizes for elements of prime power order.

  • A NEW CONSTRUCTION FOR POOLING DESIGNS

  • Part of:
  • FENGLIANG JIN, HOUCHUN ZHOU, JUAN XU
  • Published online by Cambridge University Press:
    26 September 2011, pp. 121-127
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  • Pooling designs are a very helpful tool for reducing the number of tests for DNA library screening. A disjunct matrix is usually used to represent the pooling design. In this paper, we construct a new family of disjunct matrices and prove that it has a good row to column ratio and error-tolerant property.

  • AN EXTENSION OF SURY’S IDENTITY AND RELATED CONGRUENCES

  • Part of:
  • ROMEO MEŠTROVIĆ
  • Published online by Cambridge University Press:
    04 October 2011, pp. 482-496
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  • In this paper we give an extension of a curious combinatorial identity due to B. Sury. Our proof is very simple and elementary. As an application, we obtain two congruences for Fermat quotients modulo p3. Moreover, we prove an extension of a result by H. Pan that generalizes Carlitz’s congruence.

  • JENSEN TYPE INEQUALITIES FOR Q-CLASS FUNCTIONS

  • Part of:
  • MOHAMMAD SAL MOSLEHIAN, MOHSEN KIAN
  • Published online by Cambridge University Press:
    17 October 2011, pp. 128-142
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  • Some inequalities of Jensen type for Q-class functions are proved. More precisely, a refinement of the inequality f((1/P)∑ ni=1pixi)≤P∑ ni=1(f(xi)/pi) is given in which p1,…,pn are positive numbers, P=∑ ni=1pi and f is a Q-class function. The notion of the jointly Q-class function is introduced and some Jensen type inequalities for these functions are proved. Some Ostrowski and Hermite–Hadamard type inequalities related to Q-class functions are presented as well.

  • POLAR DECOMPOSITION OF THE k-FOLD PRODUCT OF LEBESGUE MEASURE ON ℝn

  • Part of:
  • S. REZA MOGHADASI
  • Published online by Cambridge University Press:
    06 January 2012, pp. 315-324
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  • The Blaschke–Petkantschin formula is a geometric measure decomposition of the q-fold product of Lebesgue measure on ℝn. Here we discuss another decomposition called polar decomposition by considering ℝn×⋯×ℝn as ℳn×k and using its polar decomposition. This is a generalisation of the Blaschke–Petkantschin formula and may be useful when one needs to integrate a function g:ℝn×⋯×ℝn→ℝ with rotational symmetry, that is, for each orthogonal transformation O,g(O(x1),…,O(xk))=g(x1,…xk). As an application we compute the moments of a Gaussian determinant.

  • PERFECT COLOURINGS OF ISONEMAL FABRICS BY THICK STRIPING

  • Part of:
  • R. S. D. THOMAS
  • Published online by Cambridge University Press:
    12 December 2011, pp. 325-349
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  • Perfect colouring of isonemal fabrics by thick striping of warp and weft and the closely related topic of isonemal prefabrics that fall apart are reconsidered and their relation further explored. The catalogue of isonemal prefabrics of genus V that fall apart is extended to order 20 with designs that can be used to weave cubes with colour symmetry as well as weaving symmetry.

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