and Applied Analysis 3 (a, b), and let g󸀠(x) ̸ = 0 on (a, b). If f󸀠(x)/g󸀠(x) is increasing (decreasing) on (a, b), then so are f (x) − f (a) g (x) − g (a) , f (x) − f (b) g (x) − g (b) . (11) If f󸀠(x)/g󸀠(x) is strictly monotone, then the monotonicity in the conclusion is also strict. Lemma 6 (see [11, Lemma 1.1]). Suppose that the power series f(x) = ∑ ∞ n=0 a n x n and g(x) = ∑∞ n=0 b n x n hav...

In [1], Agarwal & Andrews studied partitions with "a copies of a," and in [2], Agarwal & Mullen studied partitions with "d(a) copies of a" (where d is the divisor function). In this note, partitions with "M(a) copies of a" are considered; the maximum exponent function, M, is defined by M(a) = max(e l5. .. , e T) if the integer a > 1 has canonical prime-power form a = p^ 1 ... p &r , and Af(l) =...

In a biased weak (a, b) polyform achievement game, the maker and the breaker alternately mark a, b previously unmarked cells on an infinite board, respectively. The maker’s goal is to mark a set of cells congruent to a polyform. The breaker tries to prevent the maker from achieving this goal. A winning maker strategy for the (a, b) game can be built from winning strategies for games involving f...

Consider the set S of points in the plane consisting of the ordered pairs (i, j), where 1 6 i 6 m and 1 6 j 6 n. A problem related to the study of segmentation evaluation of visual images concerns finding a permutation σ of the points of S for which the sum

1. Introducing the birank. A partition is defined as being a nonincreasing sequence of positive integers, λ = (λ1,λ2, . . . ,λr ). The set of all partitions, which includes the empty partition ∅, is denoted by . The sum of the parts of a given partition is called the weight of the partition, wt(λ) = λ1+λ2+···+λr . It is standard notation to write (z;q)∞ := ∏ t≥0(1−zq) and p−k(n) for the coeffic...

R l definable in R such that F (t) = G(t, F (t)) for all t ∈ (a, b) and each component function Gi : R 1+l → R is independent of the last l− i variables (i = 1, . . . , l). If R is o-minimal and F : (a, b) → R is R-Pfaffian, then (R, F ) is o-minimal (Proposition 7). We say that F : R → R is ultimately R-Pfaffian if there exists r ∈ R such that the restriction F ↾(r,∞) is R-Pfaffian. (In genera...

In this paper, the new concept of near PS-compactness in L-topological spaces is introduced. It is defined for any L-subset. Its characterizations and topological properties are systematically studied. And the relationship is exposed between near PScompactness and PS-compactness, and also fuzzy compactness. 2003 Elsevier Inc. All rights reserved.

Consider a linear stochastic differential equation dX(t) = (aX(t) + bX(t− 1))dt+ dW t , t ≥ 0 with time delay driven by a fractional Brownian motion {WH t , t ≥ 0}. We investigate the asymptotic properties of the maximum likelihood estimator of the parameter θ = (a, b).

In this paper we study the problem of constructing sets of s latin squares of order m such that the average number of different ordered pairs obtained by superimposing two of the s squares in the set is as large as possible. We solve this problem (for all s) when m is a prime power by using projective planes. We also present upper and lower bounds for all positive integers m.

The articles [9], [5], [11], [1], [10], [6], [3], [4], [2], [8], [12], and [7] provide the notation and terminology for this paper. We introduce metric structures which are extensions of 1-sorted structure and are systems 〈 a carrier, a distance 〉, where the carrier is a set and the distance is a function from [: the carrier, the carrier :] into R. One can check that there exists a metric struc...