Asymptotic average shadowing property on compact metric spaces

We prove that if for a continuous map f$f$ on a compact metric space X$X$, the chain recurrent set, R(f)$R\left(f\right)$ has more than one chain component, then f$f$ does not satisfy the asymptotic average shadowing property. We also show that if a continuous map f$f$ on a compact metric space X$X$ has the asymptotic average shadowing property and if A$A$ is an attractor for f$f$, then A$A$ is the single attractor for f$f$ and we have A=R(f)$A=R\left(f\right)$. We also study diffeomorphisms with asymptotic average shadowing property and prove that if M$M$ is a compact manifold which is not finite with dimM=2$dimM=2$, then the C¹$C^1$ interior of the set of all C¹$C^1$ diffeomorphisms with the asymptotic average shadowing property is characterized by the set of Ω$\Omega$-stable diffeomorphisms.     

On the exterior algebra method applied to restricted set addition

In 1994 Dias da Silva and Hamidoune solved a long-standing open problem of Erd?s and Heilbronn using the structure of cyclic spaces for derivatives on Grassmannians and the representation theory of symmetric groups. They proved that for any subset A$A$ of the p$p$-element group Z/pZ$\mathbb{Z}/p\mathbb{Z}$ (where p$p$ is a prime), at least min{p,m|A|−m²+1}$min\{p,m\left|A\right|-m^2+1\}$ different elements of the group can be written as the sum of m$m$ different elements of A$A$. In this note we present an easily accessible simplified version of their proof for the case m=2$m=2$, and explain how the method can be applied to obtain the corresponding inverse theorem.     

Crossing number additivity over edge cuts

Consider a graph G$G$ with a minimal edge cut F$F$ and let _G1$G_1$, _G2$G_2$ be the two (augmented) components of G−F$G-F$. A long-open question asks under which conditions the crossing number of G$G$ is (greater than or) equal to the sum of the crossing numbers of _G1$G_1$ and _G2$G_2$—which would allow us to consider those graphs separately. It is known that crossing number is additive for |F|∈{0,1,2}$\left|F\right|\in \{0,1,2\}$ and that there exist graphs violating this property with |F|≥4$\left|F\right|\ge 4$. In this paper, we show that crossing number is additive for |F|=3$\left|F\right|=3$, thus closing the final gap in the question.     

Zero separation results for solutions of second order linear differential equations

The oscillation of solutions of f^″+Af=0$f^{″}+Af=0$ is discussed by focusing on four separate situations. In the complex case A$A$ is assumed to be either analytic in the unit disc D$\mathbb{D}$ or entire, while in the real case A$A$ is continuous either on (−1,1)$(-1,1)$ or on (0,∞)$(0,\infty )$. In all situations A$A$ is expected to grow beyond bounds that ensure finite oscillation for all (non-trivial) solutions, and the separation between distinct zeros of solutions is considered.     

A boundary blow-up elliptic problem with an inhomogeneous term

By a perturbation method and constructing comparison functions, we reveal how the inhomogeneous term h$h$ affects the exact asymptotic behaviour of solutions near the boundary to the problem △u=b(x)g(u)+λh(x)$△u=b\left(x\right)g\left(u\right)+\lambda h\left(x\right)$, u>0$u>0$ in Ω$\Omega$, u_|∂Ω=∞$u{|}_{\partial \Omega }=\infty$, where Ω$\Omega$ is a bounded domain with smooth boundary in R^N${\mathbb{R}}^N$, λ>0$\lambda >0$, g∈C¹[0,∞)$g\in C^1[0,\infty )$ is increasing on [0,∞)$[0,\infty )$, g(0)=0$g\left(0\right)=0$, g^′$g^{\prime }$ is regularly varying at infinity with positive index ρ$\rho$, the weight b$b$, which is non-trivial and non-negative in Ω$\Omega$, may be vanishing on the boundary, and the inhomogeneous term h$h$ is non-negative in Ω$\Omega$ and may be singular on the boundary.     

A note on distance-regular graphs with a small number of vertices compared to the valency

In this note we study distance-regular graphs with a small number of vertices compared to the valency. We show that for a given α>2$\alpha >2$, there are finitely many distance-regular graphs Γ$\Gamma$ with valency k$k$, diameter D≥3$D\ge 3$ and v$v$ vertices satisfying v≤αk$v\le \alpha k$ unless (D=3$D=3$ and Γ$\Gamma$ is imprimitive) or (D=4$D=4$ and Γ$\Gamma$ is antipodal and bipartite). We also show, as a consequence of this result, that there are finitely many distance-regular graphs with valency k≥3$k\ge 3$, diameter D≥3$D\ge 3$ and _c2≥εk$c_2\ge \epsilon k$ for a given 0<ε<1$0<\epsilon <1$ unless (D=3$D=3$ and Γ$\Gamma$ is imprimitive) or (D=4$D=4$ and Γ$\Gamma$ is antipodal and bipartite).     

Embedding cycles of given length in oriented graphs

Kelly, Kühn and Osthus conjectured that for any ?≥4$?\ge 4$ and the smallest number k≥3$k\ge 3$ that does not divide ?$?$, any large enough oriented graph G$G$ with δ⁺(G),δ⁻(G)≥⌊|V(G)|/k⌋+1${\delta }^{+}\left(G\right),{\delta }^{-}\left(G\right)\ge \lfloor \left|V\left(G\right)\right|/k\rfloor +1$ contains a directed cycle of length ?$?$. We prove this conjecture asymptotically for the case when ?$?$ is large enough compared to k$k$ and k≥7$k\ge 7$. The case when k≤6$k\le 6$ was already settled asymptotically by Kelly, Kühn and Osthus.     

Modules satisfying the weak Nakayama property

Let R$R$ be a commutative ring with identity. We will say that an R$R$-module M$M$ satisfies the weak Nakayama property, if IM=M$IM=M$, where I$I$ is an ideal of R$R$, implies that for any x∈M$x\in M$ there exists a∈I$a\in I$ such that (a−1)x=0$(a-1)x=0$. In this paper, we will study modules satisfying the weak Nakayama property. It is proved that if R$R$ is a local ring, then R$R$ is a Max ring if and only if J(R)$J\left(R\right)$, the Jacobson radical of R$R$, is T$T$-nilpotent if and only if every R$R$-module satisfies the weak Nakayama property.     

On probabilistic results for the discrepancy of a hybrid-Monte Carlo sequence

In many applications it has been observed that hybrid-Monte Carlo sequences perform better than Monte Carlo and quasi-Monte Carlo sequences, especially in difficult problems. For a mixed s$s$-dimensional sequence m$m$, whose elements are vectors obtained by concatenating d$d$-dimensional vectors from a low-discrepancy sequence q$q$ with (s−d)$(s-d)$-dimensional random vectors, probabilistic upper bounds for its star discrepancy have been provided. In a paper of G. Ökten, B. Tuffin and V. Burago [G. Ökten, B. Tuffin, V. Burago, J. Complexity 22 (2006), 435–458] it was shown that for arbitrary ε>0$\epsilon >0$ the difference of the star discrepancies of the first N$N$ points of m$m$ and q$q$ is bounded by ε$\epsilon$ with probability at least 1−2exp(−ε²N/2)$1-2exp(-{\epsilon }^{2}N/2)$ for N$N$ sufficiently large. The authors did not study how large N$N$ actually has to be and if and how this actually depends on the parameters s$s$ and ε$\epsilon$. In this note we derive a lower bound for N$N$, which significantly depends on s$s$ and ε$\epsilon$. Furthermore, we provide a probabilistic bound for the difference of the star discrepancies of the first N$N$ points of m$m$ and q$q$, which holds without any restrictions on N$N$. In this sense it improves on the bound of Ökten, Tuffin and Burago and is more helpful in practice, especially for small sample sizes N$N$. We compare this bound to other known bounds.     

Counting minimal semi-Sturmian words

A finite Sturmian   word w$w$ is a balanced word over the binary alphabet {a,b}$\{a,b\}$, that is, for all subwords u$u$ and v$v$ of w$w$ of equal length, |_|u|a−_|v|a|≤1$|{\left|u\right|}_a-{\left|v\right|}_a|\le 1$, where _|u|a${\left|u\right|}_a$ and _|v|a${\left|v\right|}_a$ denote the number of occurrences of the letter a$a$ in u$u$ and v$v$, respectively. There are several other characterizations, some leading to efficient algorithms for testing whether a finite word is Sturmian. These algorithms find important applications in areas such as pattern recognition, image processing, and computer graphics. Recently, Blanchet-Sadri and Lensmire considered finite semi-Sturmian words of minimal length and provided an algorithm for generating all of them using techniques from graph theory. In this paper, we exploit their approach in order to count the number of minimal semi-Sturmian words. We also present some other results that come from applying this graph theoretical framework to subword complexity.     

On functional limits of short- and long-memory linear processes with GARCH(1,1) noises

This paper considers the short- and long-memory linear processes with GARCH (1,1) noises. The functional limit distributions of the partial sum and the sample autocovariances are derived when the tail index α$\alpha$ is in (0,2)$(0,2)$, equal to 2, and in (2,∞)$(2,\infty )$, respectively. The partial sum weakly converges to a functional of α$\alpha$-stable process when α<2$\alpha <2$ and converges to a functional of Brownian motion when α≥2$\alpha \ge 2$. When the process is of short-memory and α<4$\alpha <4$, the autocovariances converge to functionals of α/2$\alpha /2$-stable processes; and if α≥4$\alpha \ge 4$, they converge to functionals of Brownian motions. In contrast, when the process is of long-memory, depending on α$\alpha$ and β$\beta$ (the parameter that characterizes the long-memory), the autocovariances converge to either (i) functionals of α/2$\alpha /2$-stable processes; (ii) Rosenblatt processes (indexed by β$\beta$, 1/2<β<3/4$1/2<\beta <3/4$); or (iii) functionals of Brownian motions. The rates of convergence in these limits depend on both the tail index α$\alpha$ and whether or not the linear process is short- or long-memory. Our weak convergence is established on the space of càdlàg functions on [0,1]$[0,1]$ with either (i) the _J1$J_1$ or the _M1$M_1$ topology (Skorokhod, 1956); or (ii) the weaker form S$S$ topology (Jakubowski, 1997). Some statistical applications are also discussed.