On AP3-covering sequences

Recently, motivated by Stanley's sequences, Kiss, Sándor, and Yang introduced a new type sequence: a sequence A of nonnegative integers is called an $A{P}_k$-covering sequence if there exists an integer $n_0$ such that, if $n>n_0$, then there exist $a_1\in A,\dots ,a_{k?1}\in A$, $a_1<a_2<?<a_{k?1}<n$ such that $a_1,\dots ,a_{k?1},n$ form a k-term arithmetic progression. They prove that there exists an $A{P}_3$-covering sequence A such that $\underset{n\to \infty }{\lim \sup }A(n)/\sqrt{n}\le 34$. In this note, we prove that there exists an $A{P}_3$-covering sequence A such that $\underset{n\to \infty }{\lim \sup }A(n)/\sqrt{n}=\sqrt{15}$.     

Exceptional sequences of 8 line bundles on $$({\mathbb {P}}^1)^3$$ ( P 1 ) 3

Journal of Algebraic Combinatorics - We investigate maximal exceptional sequences of line bundles on $$({\mathbb {P}}^1)^r$$ , i.e., those consisting of $$2^r$$ elements. For $$r=3$$ we show that...     

A note on tilting sequences

We discuss the existence of tilting modules which are direct limits of finitely generated tilting modules over tilted algebras.      

A note on the size of the set $$\varvec{A^2+A}$$

Let \(F(x,y,z)=xy+z\). We consider some properties of expansion of the polynomial F in different settings, namely in the integers and in prime fields. The main results concern the question of covering \(\{0,1,\ldots , N\}\) (resp. \(\mathbf {F}_p\)) by \(A^2+A\) with some thin sets A.     

A note on the isotopy problem for $$ \mathbb{R}^2 $$ -planes

For a class of stable planes we define a notion of isotopy equivalence with respect to that class and prove that any two planes of a certain class of -planes comprising all affine -planes are isotopy equivalent. Furthermore we obtain that all affine -planes are isotopy equivalent in the class of affine -planes. Finally we give an example which shows that this approach cannot be easily generalized to 2-dimensional projective planes, and we outline a different way for a possible generalization.Received: 27 April 2001     

Jumping conics on a smooth quadric in $${\mathbb {P}_3}$$

We investigate the jumping conics of stable vector bundles E of rank 2 on a smooth quadric surface Q with the first Chern class c₁ = O_Q(-1,-1){c_1= \mathcal{O}_Q(-1,-1)} with respect to the ample line bundle O_Q(1,1){\mathcal {O}_Q(1,1)} . We show that the set of jumping conics of E is a hypersurface of degree c ₂(E) − 1 in \mathbb P₃^*{\mathbb {P}_3^{*}} . Using these hypersurfaces, we describe moduli spaces of stable vector bundles of rank 2 on Q in the cases of lower c ₂(E).     

A note on growth sequences of alternating groups

The aim of this paper is to give a lower bound for h(2, A_n), where h(2, A_n) is the maximum number such that A_n^{h(2, A_n)} A_n^{h(2, A_n)} can be generated by 2 elements, where A_n is the alternating group on n symbols, and n \geqq 5 n \geqq 5 . Kantor and Lubotzky (1990) gave a lower bound¶ \fracn!8 \frac{n!}{8} for sufficiently large n by the probability of generating the symmetric group. I have improved the above lower bound to \fracn!5 \frac{n!}{5} for large n, using a different method.     

A note on the hardness of Skolem-type sequences

The purpose of this note is to give upper bounds (assuming different from ) on how far the generalizations of Skolem sequences can be taken while still hoping to resolve the existence question. We prove that the existence questions for both multi-Skolem sequences and generalized Skolem sequences are strongly -complete. These results are significant strengthenings and simplifications of the recent -completeness result for generalized multi-Skolem sequences.     

A note on the nonexistence of Barker sequences

A Barker sequence is a sequence with elements ±1 such that all out-of-phase aperiodic autocorrelation coefficients are 0, 1 or -1. It is known that if a Barker sequence of length s > 13 exists then s = 4N ² for some odd integer N 55, and it has long been conjectured that no such sequence exists. We review some previous attempts to improve the bound on N which, unfortunately, contain errors. We show that a recent theorem of Eliahou et al. [5] rules out all but six values of N less than 5000, the smallest of which is 689.     

A note on shifted convolution of cusp-forms with $$\theta $$-series

Recently Lachterman, Schayer, and Younger published an elegant proof of the Ramanujan congruences for the partition function p(n). Their proof uses only the classical theory of modular forms as well as a beautiful result of Choie, Kohnen, and Ono, without the need for Hecke operators. In this paper, we give a method for generalizing Lachterman, Schayer, and Younger’s proof to include Ramanujan congruences for multipartition functions \(p_k(n)\) and Ramanujan congruences for p(n) modulo certain prime powers.     

On the pseudorandomness of quaternary sequences derived from sequences over $$\mathbb F_4$$

In analogy to the corresponding measures of pseudorandomness for quaternary sequences introduced by Mauduit and Sárközy (for m-ary sequences) we introduce the well-distribution measure and correlation measure of order k for sequences over \(\mathbb F_4\). Using any fixed bijection from \(\mathbb F_4\) to the set of complex fourth roots of unity, we analyze the relation of these pseudorandomness measures for sequences over \(\mathbb F_4\) and for the corresponding quaternary sequences. More precisely, we show that they differ only by a multiplicative constant (depending only on k). We also apply the results for deriving new quaternary pseudorandom sequences from pseudorandom sequences over \(\mathbb F_4\) and vice versa.     

A note on fractional $${\varvec{}}$$-Lalacian roblems with singular weights

We study a class of fractional p-Laplacian problems with weights which are possibly singular on the boundary of the domain. We provide existence and multiplicity results as well as characterizations of critical groups and related applications.     

A note on generating functions for Hausdorff moment sequences

For functions whose Taylor coefficients at the origin form a Hausdorff moment sequence we study the behaviour of for ( fixed).