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.     

Cycles embedding on folded hypercubes with faulty nodes

Let F_Fv$F{F}_v$ be the set of faulty nodes in an n$n$-dimensional folded hypercube F_Qn$F{Q}_n$ with |F_Fv|≤n−2$\left|F{F}_v\right|\le n-2$. In this paper, we show that if n≥3$n\ge 3$, then every edge of F_Qn−F_Fv$F{Q}_n-F{F}_v$ lies on a fault-free cycle of every even length from 4$4$ to 2ⁿ−2|F_Fv|$2^n-2\left|F{F}_v\right|$, and if n≥2$n\ge 2$ and n$n$ is even, then every edge of F_Qn−F_Fv$F{Q}_n-F{F}_v$ lies on a fault-free cycle of every odd length from n+1$n+1$ to 2ⁿ−2|F_Fv|−1$2^n-2\left|F{F}_v\right|-1$.     

On weighted Hilbert spaces and integration of functions of infinitely many variables

We study aspects of the analytic foundations of integration and closely related problems for functions of infinitely many variables _x1,_x2,…∈D$x_1,x_2,\dots \in D$. The setting is based on a reproducing kernel k$k$ for functions on D$D$, a family of non-negative weights _γu${\gamma }_u$, where u$u$ varies over all finite subsets of N$\mathbb{N}$, and a probability measure ρ$\rho$ on D$D$. We consider the weighted superposition K=_∑uγuku$K={\sum }_u{\gamma }_u{k}_u$ of finite tensor products _ku$k_u$ of k$k$. Under mild assumptions we show that K$K$ is a reproducing kernel on a properly chosen domain in the sequence space D^N$D^{\mathbb{N}}$, and that the reproducing kernel Hilbert space H(K)$H\left(K\right)$ is the orthogonal sum of the spaces H(_γuku)$H\left({\gamma }_u{k}_u\right)$. Integration on H(K)$H\left(K\right)$ can be defined in two ways, via a canonical representer or with respect to the product measure ρ^N${\rho }^{\mathbb{N}}$ on D^N$D^{\mathbb{N}}$. We relate both approaches and provide sufficient conditions for the two approaches to coincide.     

Critical exponents for non-simultaneous blow-up in a localized parabolic system

The paper deals with the radially symmetric solutions of _ut=Δu+u^m(x,t)vⁿ(0,t)$u_t=\Delta u+u^m(x,t)v^n(0,t)$, _vt=Δv+u^p(0,t)v^q(x,t)$v_t=\Delta v+u^p(0,t)v^q(x,t)$, subject to null Dirichlet boundary conditions. For the blow-up classical solutions, we propose the critical exponents for non-simultaneous blow-up by determining the complete and optimal classification for all the non-negative exponents: (i) There exist initial data such that u$u$ (v$v$) blows up alone if and only if m>p+1$m>p+1$ (q>n+1$q>n+1$), which means that any blow-up is simultaneous if and only if m≤p+1$m\le p+1$, q≤n+1$q\le n+1$. (ii) Any blow-up is u$u$ (v$v$) blowing up with v$v$ (u$u$) remaining bounded if and only if m>p+1$m>p+1$, q≤n+1$q\le n+1$ (m≤p+1$m\le p+1$, q>n+1$q>n+1$). (iii) Both non-simultaneous and simultaneous blow-up may occur if and only if m>p+1$m>p+1$, q>n+1$q>n+1$. Moreover, we consider the blow-up rate and set estimates which were not obtained in the previously known work for the same model.     

Unbounded order convergence in dual spaces

A net (_xα)$(x_{\alpha })$ in a vector lattice X   is said to be unbounded order convergent (or uo-convergent, for short) to x∈X$x\in X$ if the net (|_xα−x|∧y)$(|x_{\alpha }-x|\wedge y)$ converges to 0 in order for all y∈_X+$y\in X_{+}$. In this paper, we study unbounded order convergence in dual spaces of Banach lattices. Let X   be a Banach lattice. We prove that every norm bounded uo-convergent net in X^?$X^{?}$ is w^?$w^{?}$-convergent iff X   has order continuous norm, and that every w^?$w^{?}$-convergent net in X^?$X^{?}$ is uo-convergent iff X is atomic with order continuous norm. We also characterize among σ  -order complete Banach lattices the spaces in whose dual space every simultaneously uo- and w^?$w^{?}$-convergent sequence converges weakly/in norm.     

Blow-up and propagation of disturbances for fast diffusion equations

This paper is concerned with the Cauchy problem for the fast diffusion equation _ut−Δu^m=αu^_p1$u_t-\Delta u^m=\alpha u^{p_1}$ in R^N${\mathbb{R}}^N$ (N≥1$N\ge 1$), where m∈(0,1)$m\in (0,1)$, _p1>1$p_1>1$ and α>0$\alpha >0$. The initial condition _u0$u_0$ is assumed to be continuous, nonnegative and bounded. Using a technique of subsolutions, we set up sufficient conditions on the initial value _u0$u_0$ so that u(t,x)$u(t,x)$ blows up in finite time, and we show how to get estimates on the profile of u(t,x)$u(t,x)$ for small enough values of t>0$t>0$.     

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.     

On first-fit coloring of ladder-free posets

Bosek and Krawczyk exhibited an on-line algorithm for partitioning an on-line poset of width w$w$ into w^14lgw$w^{14lgw}$ chains. They also observed that the problem of on-line chain partitioning of general posets of width w$w$ could be reduced to First-Fit chain partitioning of 2w²+1$2w^2+1$-ladder-free posets of width w$w$, where an m$m$-ladder is the transitive closure of the union of two incomparable chains _x1≤?≤_xm$x_1\le ?\le x_m$, _y1≤?≤_ym$y_1\le ?\le y_m$ and the set of comparabilities {_x1≤_y1,…,_xm≤_ym}$\{x_1\le y_1,\dots ,x_m\le y_m\}$. Here, we provide a subexponential upper bound (in terms of w$w$ with m$m$ fixed) for the performance of First-Fit chain partitioning on m$m$-ladder-free posets, as well as an exact quadratic bound when m=2$m=2$, and an upper bound linear in m$m$ when w=2$w=2$. Using the Bosek–Krawczyk observation, this yields an on-line chain partitioning algorithm with a somewhat improved performance bound. More importantly, the algorithm and the proof of its performance bound are much simpler.     

On words that are concise in residually finite groups

A group-word w$w$ is called concise if whenever the set of w$w$-values in a group G$G$ is finite it always follows that the verbal subgroup w(G)$w\left(G\right)$ is finite. More generally, a word w$w$ is said to be concise in a class of groups X$X$ if whenever the set of w$w$-values is finite for a group G∈X$G\in X$, it always follows that w(G)$w\left(G\right)$ is finite. P. Hall asked whether every word is concise. Due to Ivanov the answer to this problem is known to be negative. Dan Segal asked whether every word is concise in the class of residually finite groups. In this direction we prove that if w$w$ is a multilinear commutator and q$q$ is a prime-power, then the word w^q$w^q$ is indeed concise in the class of residually finite groups. Further, we show that in the case where w=_γk$w={\gamma }_k$ the word w^q$w^q$ is boundedly concise in the class of residually finite groups. It remains unknown whether the word w^q$w^q$ is actually concise in the class of all groups.     

Uniqueness properties of m-subharmonic functions in Cegrell classes

Let u,v$u,v$ be m-subharmonic functions defined on a domain Ω   in Cⁿ${\mathbb{C}}^n$. We are interested in giving sufficient conditions on u,v$u,v$ such that u=v$u=v$ on the whole domain Ω. Some applications to weak convergence of sequence of m-subharmonic functions are also discussed.     

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.     

The Deodhar decomposition of the Grassmannian and the regularity of KP solitons

Given a point A$A$ in the real Grassmannian, it is well-known that one can construct a soliton solution _uA(x,y,t)$u_A(x,y,t)$ to the KP equation. The contour plot   of such a solution provides a tropical approximation to the solution when the variables x$x$, y$y$, and t$t$ are considered on a large scale and the time t$t$ is fixed. In this paper we use several decompositions of the Grassmannian in order to gain an understanding of the contour plots of the corresponding soliton solutions. First we use the positroid stratification   of the real Grassmannian in order to characterize the unbounded line-solitons in the contour plots at y?0$y?0$ and y?0$y?0$. Next we use the Deodhar decomposition   of the Grassmannian–a refinement of the positroid stratification–to study contour plots at t?0$t?0$. More specifically, we index the components of the Deodhar decomposition of the Grassmannian by certain tableaux which we call Go-diagrams  , and then use these Go-diagrams to characterize the contour plots of solitons solutions when t?0$t?0$. Finally we use these results to show that a soliton solution _uA(x,y,t)$u_A(x,y,t)$ is regular for all times t$t$ if and only if A$A$ comes from the totally non-negative part of the Grassmannian.     

Boundedness of extremal solutions in dimension 4

In this paper we establish the boundedness of the extremal solution u^∗$u^{\ast }$ in dimension N=4$N=4$ of the semilinear elliptic equation −Δu=λf(u)$-\Delta u=\lambda f\left(u\right)$, in a general smooth bounded domain Ω⊂R^N$\Omega \subset {\mathbb{R}}^N$, with Dirichlet data u_|∂Ω=0$u{|}_{\partial \Omega }=0$, where f$f$ is a C¹$C^1$ positive, nondecreasing and convex function in [0,∞)$[0,\infty )$ such that f(s)/s→∞$f\left(s\right)/s\to \infty$ as s→∞$s\to \infty$.     

A Sobolev inequality with reciprocal weights

We give a Sobolev inequality with the weight K(x)$K\left(x\right)$ belonging to the class _A2∩_Gn$A_2\cap G_n$ for the function |u|^t${\left|u\right|}^t$ and the weight K(x)⁻¹$K{\left(x\right)}^{-1}$ for |∇u|²${\left|\nabla u\right|}^2$. The constant in the relevant inequality is seen to depend on the _Gn$G_n$ and _A2$A_2$ constants of the weight.     

Zeros of the Macdonald function of complex order

The z  -zeros of the modified Bessel function of the third kind _Kν(z)$K_{\nu }(z)$, also known as modified Hankel function or Macdonald function, are considered for arbitrary complex values of the order ν$\nu$. Approximate expressions for the zeros, applicable in the cases of very small or very large |ν|$\left|\nu \right|$, are given. The behaviour of the zeros for varying |ν|$\left|\nu \right|$ or argν$\arg \nu$, obtained numerically, is illustrated by means of some graphics.     

The regularity of generalized solutions for the n-dimensional quasi-linear parabolic diffraction problems

In this paper, we study the regularity of generalized solutions u(x,t)$u(x,t)$ for the n  -dimensional quasi-linear parabolic diffraction problem. By using various estimates and Steklov average methods, we prove that (1): for almost all t$t$ the first derivatives _ux(x,t)$u_x(x,t)$ are Hölder continuous with respect to x$x$ up to the inner boundary, on which the coefficients of the equation are allowed to be discontinuous; and (2): the first derivative _ut(x,t)$u_t(x,t)$ is Hölder continuous with respect to (x,t)$(x,t)$ across the inner boundary.