The index of small length sequences

Let [Formula: see text] be a fixed integer. Define [Formula: see text] to be the unique integer in the range [Formula: see text] which is congruent to [Formula: see text] modulo [Formula: see text]. Given [Formula: see text], let [Formula: see text] and define [Formula: see text] to be the index of the sequence [Formula: see text]. If [Formula: see text] have [Formula: see text] but [Formula: see text] for all proper, non-empty subsets [Formula: see text], then a still open conjecture asserts that [Formula: see text] provided that [Formula: see text]. We give an alternative proof, that does not rely on computer calculations, verifying this conjecture when [Formula: see text] is a product of two prime powers.

Tridiagonal pairs of type III with height one

Let K denote an algebraically closed field with characteristic 0. Let V denote a vector space over K with finite positive dimension, and let A, A∗ denote a tridiagonal pair on V of diameter d. Let V0, . . . , Vd denote a standard ordering of the eigenspaces of A on V , and let θ0, . . . , θd denote the corresponding eigenvalues of A. It is assumed that d ≥ 3. Let ρi denote the dimension of Vi. The sequence ρ0, ρ1, . . . , ρd is called the shape of the tridiagonal pair. It is known that ρ0 = 1 and there exists a unique integer h (0 ≤ h ≤ d/2) such that ρi−1 < ρi for 1 ≤ i ≤ h, ρi−1 = ρi for h < i ≤ d − h, and ρi−1 > ρi for d − h < i ≤ d. The integer h is known as the height of the tridiagonal pair. In this paper, it is showed that the shape of a tridiagonal pair of type III with height one is either 1, 2, 2, . . ., 2, 1 or 1, 3, 3, 1. In each case, an interesting basis is found for V and the actions of A, A∗ on this basis are described.

Chover-Type Laws of the Iterated Logarithm for Kesten-Spitzer Random Walks in Random Sceneries Belonging to the Domain of Stable Attraction

Let X={Xi,i≥1} be a sequence of real valued random variables, S0=0 and Sk=∑i=1kXi (k≥1). Let σ={σ(x),x∈Z} be a sequence of real valued random variables which are independent of X’s. Denote by Kn=∑k=0nσ(⌊Sk⌋) (n≥0) Kesten-Spitzer random walk in random scenery, where ⌊a⌋ means the unique integer satisfying ⌊a⌋≤a<⌊a⌋+1. It is assumed that σ’s belong to the domain of attraction of a stable law with index 0<β<2. In this paper, by employing conditional argument, we investigate large deviation inequalities, some sufficient conditions for Chover-type laws of the iterated logarithm and the cluster set for random walk in random scenery Kn. The obtained results supplement to some corresponding results in the literature.

On the zeros of linear combinations of derivatives of the Riemann zeta function, II

The relevant number to the Dirichlet series [Formula: see text], is defined to be the unique integer [Formula: see text] with [Formula: see text], which maximizes the quantity [Formula: see text]. In this paper, we classify the set of all relevant numbers to the Dirichlet [Formula: see text]-functions. The zeros of linear combinations of [Formula: see text] and its derivatives are also studied. We give an asymptotic formula for the supremum of the real parts of zeros of such combinations. We also compute the degree of the largest derivative needed for such a combination to vanish at a certain point.

SCHEDULING WITH POSITION-BASED DETERIORATING JOBS AND MULTIPLE DETERIORATING RATE-MODIFYING ACTIVITIES

In a recent paper, Ozturkoglu and Bulfin (Ozturkoglu, Y. and RL Bulfin (2011). A unique integer mathematical model for scheduling deteriorating jobs with rate-modifying activities on a single machine. The International Journal of Advanced Manufacturing Technology, 57, 753–762.) formulate a unique integer program to solve the single-machine scheduling for the objectives of minimizing makespan and total completion time. They also propose efficient heuristic algorithms for solving large size problems. However their heuristics are not optimal and so the NP-hardness of the considered problem is still open. In this note, we show that a more general problem can be optimally solved in polynomial time. We also provide optimal polynomial-time solution algorithm for the parallel-machine case to minimize total completion time.

THE SYMMETRIC CROSSCAP NUMBER OF THE GROUPS OF SMALL-ORDER

Every finite group G acts as an automorphism group of some non-orientable Klein surfaces without boundary. The minimal genus of these surfaces is called the symmetric crosscap number and denoted by [Formula: see text]. It is known that 3 cannot be the symmetric crosscap number of a group. Conversely, it is also known that all integers that do not belong to nine classes modulo 144 are the symmetric crosscap number of some group. Here we obtain infinitely many groups whose symmetric crosscap number belong to each one of six of these classes. This result supports the conjecture that 3 is the unique integer which is not the symmetric crosscap number of a group. On the other hand, there are infinitely many groups with symmetric crosscap number 1 or 2. For g > 2 the number of groups G with [Formula: see text] is finite. The value of [Formula: see text] is known when G belongs to certain families of groups. In particular, if o(G) < 32, [Formula: see text] is known for all except thirteen groups. In this work we obtain it for these groups by means of a one-by-one analysis. Finally we obtain the least genus greater than two for those exceptional groups whose symmetric crosscap number is 1 or 2.