Discrete Tomography and plane partitions

A plane partition   is a p×q$p\times q$ matrix A=(_aij)$A=(a_{ij})$, where 1?i?p$1?i?p$ and 1?j?q$1?j?q$, with non-negative integer entries, and whose rows and columns are weakly decreasing. From a geometric point of view plane partitions are equivalent to pyramids  , subsets of the integer lattice Z³${\mathbb{Z}}^3$ which play an important role in Discrete Tomography. As a consequence, some typical problems concerning the tomography of discrete lattice sets can be rephrased and considered via plane partitions. In this paper we focus on some of them. In particular, we get a necessary and sufficient condition for additivity, a canonical procedure for checking the existence of (weakly) bad configurations, and an algorithm which constructs minimal pyramids (with respect to the number of levels) with assigned projection of a bad configurations.     

Muckenhoupt weights and Lindelöf theorem for harmonic mappings

We extend the result of Lavrentiev which asserts that the harmonic measure and the arc-length measure are _A∞$A_{\infty }$ equivalent in a chord-arc Jordan domain. By using this result we extend the classical result of Lindelöf to the class of quasiconformal (q.c.) harmonic mappings by proving the following assertion. Assume that f is a quasiconformal harmonic mapping of the unit disk U onto a Jordan domain. Then the function A(z)=arg?(_∂φ(f(z))/z)$A(z)=\arg ?({\partial }_{\phi }(f(z))/z)$ where z=re^iφ$z=r{e}^{i\phi }$, is well-defined and smooth in U^?={z:0<|z|<1}${\mathbf{U}}^{?}=\{z:0<|z|<1\}$ and has a continuous extension to the boundary of the unit disk if and only if the image domain has C¹$C^1$ boundary.     

Oscillating rim hook tableaux and colored matchings

We find a correspondence between oscillating m-rim hook tableaux and m-colored matchings, where m is a positive integer. An oscillating m  -rim hook tableau is defined as a sequence (λ⁰,λ¹,…,λ2n)$({\lambda }^0,{\lambda }^1,\dots ,{\lambda }^{2n})$ of Young diagrams starting with the empty shape and ending with the empty shape such that λi${\lambda }^i$ is obtained from λi−1${\lambda }^{i-1}$ by adding an m-rim hook or by deleting an m-rim hook. Our bijection relies on the generalized Schensted algorithm due to White. An oscillating 2-rim hook tableau is also called an oscillating domino tableau. When we restrict our attention to two column oscillating domino tableaux of length 2n  , we are led to a bijection between such tableaux and noncrossing 2-colored matchings on {1,2,…,2n}$\{1,2,\dots ,2n\}$, which are counted by the product CnC_n+1$C_n{C}_{n+1}$ of two consecutive Catalan numbers. A 2-colored matching is noncrossing if there are no two arcs of the same color that are intersecting. We show that oscillating domino tableaux with at most two columns are in one-to-one correspondence with Dyck path packings. A Dyck path packing of length 2n   is a pair (D,E)$(D,E)$, where D is a Dyck path of length 2n, and E is a dispersed Dyck path of length 2n that is weakly covered by D. So we deduce that Dyck path packings of length 2n   are counted by CnC_n+1$C_n{C}_{n+1}$.     

On fractal properties of non-normal numbers with respect to Rényi f-expansions generated by piecewise linear functions

The paper is devoted to the study of fractal properties of subsets of the set of non-normal numbers with respect to Rényi f  -expansions generated by continuous increasing piecewise linear functions defined on [0,+∞)$[0,+\infty )$. All such expansions are expansions for real numbers generated by infinite linear IFS f={_f0,_f1,…,_fn,…}$f=\{f_0,f_1,\dots ,f_n,\dots \}$ with the following list of ratios _Q∞=(_q0,_q1,…,_qn,…)$Q_{\infty }=(q_0,q_1,\dots ,q_n,\dots )$.     

Log-convexity and strong q-log-convexity for some triangular arrays

We give a criterion for the log-convexity (resp. the strong q  -log-convexity) of the first column of certain infinite triangular array _(An,k)0?k?n${(A_{n,k})}_{0?k?n}$ of nonnegative numbers (resp. of polynomials in q with nonnegative coefficients), for which the recurrence relation is of the form

_An,k=_{fkAn−1,k−1}+_gkAn−1,k+_hkAn−1,k+1.$A_{n,k}=f_k{A}_{n-1,k-1}+g_k{A}_{n-1,k}+h_k{A}_{n-1,k+1}.$

This allows a unified treatment of the log-convexity of the Catalan-like numbers, as well as that of the q-log-convexity of some classical polynomials. In particular, we obtain simple proofs of the q-log-convexity of Narayana polynomials.     

Ramsey Theory,integer partitions and a new proof of the Erdős–Szekeres Theorem

Let H be a k  -uniform hypergraph whose vertices are the integers 1,…,N$1,\dots ,N$. We say that H contains a monotone path of length n   if there are _x1<_x2<?<_xn+k−1$x_1<x_2<?<x_{n+k-1}$ so that H contains all n   edges of the form {_xi,_xi+1,…,_xi+k−1}$\{x_i,x_{i+1},\dots ,x_{i+k-1}\}$. Let _Nk(q,n)$N_k(q,n)$ be the smallest integer N so that every q-coloring of the edges of the complete k-uniform hypergraph on N vertices contains a monochromatic monotone path of length n  . While the study of _Nk(q,n)$N_k(q,n)$ for specific values of k and q   goes back (implicitly) to the seminal 1935 paper of Erd?s and Szekeres, the problem of bounding _Nk(q,n)$N_k(q,n)$ for arbitrary k and q was studied by Fox, Pach, Sudakov and Suk.     

Circulant partial Hadamard matrices

A question arising in stream cypher cryptanalysis is reframed and generalized in the setting of Hadamard matrices as follows: For given n, what is the maximum value of k   for which there exists a k×n$k\times n$(±1)$(\pm 1)$-matrix A   such that AA^T=n_Ik$A{A}^{\mathsf{T}}=n{I}_k$, with each row after the first obtained by a cyclic shift of its predecessor by one position? For obvious reasons we call such matrices circulant partial Hadamard matrices. Further, what is the maximum value of k subject to the condition that the row sums are equal to r?     

On oriented graphs whose skew spectral radii do not exceed 2

Let S(G^σ)$S(G^{\sigma })$ be the skew adjacency matrix of the oriented graph G^σ$G^{\sigma }$ of order n   and _λ1,_λ2,…,_λn${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ be all eigenvalues of S(G^σ)$S(G^{\sigma })$. The skew spectral radius _ρs(G^σ)${\rho }_s(G^{\sigma })$ of G^σ$G^{\sigma }$ is defined as max{|_λ1|,|_λ2|,…,|_λn|}$\max \{|{\lambda }_1|,|{\lambda }_2|,\dots ,|{\lambda }_n|\}$. In this paper, we investigate oriented graphs whose skew spectral radii do not exceed 2.     

THE SHARP ESTIMATE OF THE THIRD HOMOGENEOUS EXPANSION FOR A CLASS OF STARLIKE MAPPINGS OF ORDER α ON THE UNIT POLYDISK IN Cn

In this article,first,a sufficient condition for a starlike mapping of order α f(x) defined on the unit ball in a complex Banach space is given. Second,the sharp estimate of the third homogeneous expan...     

Effective Pólya semi-positivity for non-negative polynomials on the simplex

We consider homogeneous polynomials f∈R[_x1,…,_xn]$f\in \mathbb{R}[x_1,\dots ,x_n]$ which are non-negative on the standard simplex in Rⁿ${\mathbb{R}}^n$, and we obtain sufficient conditions for such an f   to be Pólya semi-positive, that is, all the coefficients of (_x1+?+_xn)^Nf$(x_1+?+x_n{)}^{N}f$ are non-negative for all sufficiently large positive integers N. Such sufficient conditions are expressed in terms of the vanishing orders of the monomial terms of f along the faces of the simplex. Our result also gives effective estimates on N   under such conditions. Moreover, we also show that any Pólya semi-positive polynomial necessarily satisfies a slightly weaker condition. In particular, our results lead to a simple characterization of the Pólya semi-positive polynomials in the low dimensional case when n?3$n?3$ as well as the case (in any dimension) when the zero set of the polynomial in the simplex consists of a finite number of points. We also discuss an application to the representations of non-homogeneous polynomials which are non-negative on a general simplex.