On the extended Hardy-Hilbert's inequality

In this paper, by introducing three parameters A, B, and λ, and estimating the weight coefficient, we give a generalization of the extended Hardy-Hilbert's inequality with a best possible constant factor, involving the β function. We also consider its equivalent inequality and the associated double series form.     

On the extended Hilbert's inequality

In this paper, it is shown that the extended Hilbert's inequality for double series can be refined by the aid of the Euler-Maclaurin summation formula. The extreme cases and are discussed.

     

On the extended tube conjecture

We present a new approach to the extended tube conjecture. Research supported by KBN Grant No 2 PO3A 060 08 and by Stiftung für deutschpolnische Zusammenarbeit 632/93     

On the extended linear complementarity problem

For the extended linear complementarity problem (Mangasarian and Pang, 1995), we introduce and characterize column-sufficiency, row-sufficiency andP-properties. These properties are then specialized to the vertical, horizontal and mixed linear complementarity problems. This paper is dedicated to Professor Olvi L. Mangasarian on the occasion of his 60th birthday.     

On the genus of elliptic fibrations

A simply connected topological space is called elliptic if both and are finite-dimensional -vector spaces. In this paper, we consider fibrations for which the fibre is elliptic and is evenly graded. We show that in the generic cases, the genus of such a fibration is completely determined by generalized Chern classes of the fibration.

     

On extended algebraic immunity

Algebraic immunity (AI) measures the resistance of a Boolean function f against algebraic attack. Extended algebraic immunity (EAI) extends the concept of algebraic immunity, whose point is that a Boolean function f may be replaced by another Boolean function f ^c called the algebraic complement of f. In this paper, we study the relation between different properties (such as weight, nonlinearity, etc.) of Boolean function f and its algebraic complement f ^c . For example, the relation between annihilator sets of f and f ^c provides a faster way to find their annihilators than previous report. Next, we present a necessary condition for Boolean functions to be of the maximum possible extended algebraic immunity. We also analyze some Boolean functions with maximum possible algebraic immunity constructed by known existing construction methods for their extended algebraic immunity.     

On the genus and thickness of graphs

The genus γ(G) of a simple graph G is the minimum genus of the orientable surface on which G is embeddable. The thickness θ(G) of G is the minimum number of planar subgraphs of G whose union is G. From the definitions, it is clear that θ(G) = 1 if and only if γ(G) = 0. In this paper, we will show that θ(G) ≦ γ(G) + 1, if G has no triangle or if G is toroidal.     

On the average genus of a graph

Not all rational numbers are possibilities for the average genus of an individual graph. The smallest such numbers are determined, and varied examples are constructed to demonstrate that a single value of average genus can be shared by arbitrarily many different graphs. It is proved that the number 1 is a limit point of the set of possible values for average genus and that the complete graph K₄ is the only 3-connected graph whose average genus is less than 1.     

On globally solving the extended trust-region subproblems

With the help of the newly developed technique—second order cone (SOC) constraints to strengthen the SDP relaxation of the extended trust-region subproblem (eTRS), we modify two recent SDP relaxation based branch and bound algorithms for solving eTRS. Numerical experiments on some types of problems show that the new algorithms run faster for finding the global optimal solutions than the SDP relaxation based algorithms.     

On the genus of a maximal curve

The upper limit and the first gap in the spectrum of genera of -maximal curves are known, see [34], [16], [35]. In this paper we determine the second gap. Both the first and second gaps are approximately constant times , but this does not hold true for the third gap which is just 1 for while (at most) constant times q for This suggests that the problem of determining the third gap which is the object of current work on -maximal curves could be intricate. Here, we investigate a relevant related problem namely that of characterising those -maximal curves whose genus is equal to the third (or possible the forth) largest value in the spectrum. Our results also provide some new evidence on -maximal curves in connection with Castelnuovo's genus bound, Halphen's theorem, and extremal curves. Received: 1 January 2001 / Revised version: 30 July 2001 / Published online: 23 May 2002     

On the genus of groups with operators

Charles Cassidy has given a definition of genus for nilpotent groups with operators, generalizing the Mislin genus. Here we modify Cassidy's definition to bring it into line with homotopy theory. Thus, with our definition, if X and Y are nilpotent spaces in the same genus, then the π₁X-group π_nX is in the genus of the π₁Y-group π_nY. Relations are found between the genus of the Q-group N and the (Mislin) genus of the semidirect product N

Q.     

On the genus of a random graph

Let p = p(n) be a function of n with 0<p<1. We consider the random graph model ??(n, p); that is, the probability space of simple graphs with vertex-set {1, 2,…, n}, where two distinct vertices are adjacent with probability p. and for distinct pairs these events are mutually independent. Archdeacon and Grable have shown that if p²(1 ? p²) ?? 8(log n)⁴/n. then the (orientable) genus of a random graph in ??(n, p) is (1 + o(1))pn²/12. We prove that for every integer i ? 1, if n^?i/(i + 1) «p «n^{?(i ? 1)/i}. then the genus of a random graph in ??(n, p) is (1 + o(1))i/4(i + 2) pn². If p = cn^?(i?1)/o, where c is a constant, then the genus of a random graph in ??(n, p) is (1 + o(1))g(i, c, n)pn² for some function g(i, c, n) with 1/12 ? g(i, c, n) ? 1. but for i > 1 we were unable to compute this function.     

On the minimal genus of 2-complexes

Gross and Rosen asked if the genus of a 2-dimensional complex K embeddable in some (orientable) surface is equal to the genus of the graph of appropriate barycentric subdivision of K. We answer the nonorientable genus and the Euler genus versions of Gross and Rosen's question in affirmative. We show that this is not the case for the orientable genus by proving that taking ⌊ log₂ g⌋ th barycentric subdivision is not sufficient, where g is the genus of K. On the other hand, (1+⌈log₂(g+2)⌉)th subdivision is proved to be sufficient. © 1997 John Wiley & Sons, Inc.