Comments on the height reducing property

A complex number α is said to satisfy the height reducing property if there is a finite subset, say F, of the ring ? of the rational integers such that ?[α] = F[α]. This property has been considered by several authors, especially in contexts related to self affine tilings and expansions of real numbers in non-integer bases. We prove that a number satisfying the height reducing property, is an algebraic number whose conjugates, over the field of the rationals, are all of modulus one, or all of modulus greater than one. Expecting the converse of the last statement is true, we show some theoretical and experimental results, which support this conjecture.     

Application of X-ray fluorescence spectroscopy to provenance studies of Algerian archaeological pottery

The North African regions boast a multimillenary tradition of pottery production that represents a decisive progress in all societies. In this paper, we present the analysis of several fragments of pottery dating from the Middle Ages and collected in the archaeological site of Gouraya in Béjaïa (Algeria). Energy-dispersive X-ray fluorescence measurements were performed to obtain the elemental composition of the clays and to attempt a classification of the samples through multivariate analysis methods, connecting them to the archaeological information about the sites. Hierarchical clusters analysis (HCA) identified two possible different groups, related to different production sites, even if only one source of clay seem to be used, consistently with the geological contest. Based on the results, a tentative provenance interpretation could be proposed on historical bases. We put in place a strategy that made it possible to perform provenance studies respecting sample integrity as we aim to extend our research also to integer and well-preserved object from museum, to create a database for Algerian production of in the Middle Ages.     

Partitions of a set satisfying certain set of conditions

In this paper we present a new combinatorial class enumerated by Catalan numbers. More precisely, we establish a bijection between the set of partitions π₁π₂?π_n of [n] such that π_i+1−π_i≤1 for all i=,1,2…,n−1, and the set of Dyck paths of semilength n. Moreover, we find an explicit formula for the generating function for the number of partitions π₁π₂?π_n of [n] such that either π_i+?−π_i≤1 for all i=1,2,…,n−?, or π_i+1−π_i≤m for all i=1,2,…,n−1.     

Restricted 132-Alternating Permutations and Chebyshev Polynomials

A permutation is said to be alternating if it starts with rise and then descents and rises come in turn. In this paper we study the generating function for the number of alternating permutations on n letters that avoid or contain exactly once 132 and also avoid or contain exactly once an arbitrary pattern on k letters. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind.AMS Subject Classification: 05A05, 05A15, 30B70, 42C05.     

Average norms of polynomials

We present an explicit formula for the average -norm over all the polynomials of degree n with coefficients in T, where T is a finite set of complex numbers and α is a positive integer.     

Approximate Controllability of Semilinear Fractional Stochastic Dynamic Systems with Nonlocal Conditions in Hilbert Spaces

A class of dynamic control systems described by semilinear fractional stochastic differential equations of order 1 < q < 2 with nonlocal conditions in Hilbert spaces is considered. Using solution operator theory, fractional calculations, fixed-point technique and methods adopted directly from deterministic control problems, a new set of sufficient conditions for nonlocal approximate controllability of semilinear fractional stochastic dynamic systems is formulated and proved by assuming the associated linear system is approximately controllable. As a remark, the conditions for the exact controllability results are obtained. Finally, an example is provided to illustrate the obtained theory.     

The perimeter of words

We define $\left[k\right]=\{1,2,\dots ,k\}$ to be a (totally ordered) alphabet on $k$ letters. A word $w$ of length $n$ on the alphabet $\left[k\right]$ is an element of ${\left[k\right]}^n$. A word can be represented by a bargraph (i.e., by a column-convex polyomino whose lower edges lie on the $x$-axis) in which the height of the $i$th column equals the size of the $i$th part of the word. Thus these bargraphs have heights which are less than or equal to $k$. We consider the perimeter, which is the number of edges on the boundary of the bargraph. By way of Cramer’s method and the kernel method, we obtain the generating function that counts the perimeter of words. Using these generating functions we find the average perimeter of words of length $n$ over the alphabet $\left[k\right]$. We also show how the mean and variance can be obtained using a direct counting method.     

Bargraph statistics on words and set partitions

In this paper, we consider statistics on partitions of an n-element set represented as a subset of the bargraphs that have n horizontal steps. More precisely, we find the joint distribution of the area and up step statistics on the latter subset of bargraphs, thereby obtaining new refined counts on partitions having a fixed number of blocks. Furthermore, we give explicit formulas in terms of the Stirling numbers for the total area and number of up steps in bargraphs corresponding to partitions, providing both algebraic and combinatorial proofs. Finally, we find asymptotic estimates for the average and total values of these statistics and as a consequence obtain some new identities for the Stirling and Bell numbers.     

Photocatalytic activity of silicon nanowires under UV and visible light irradiation

The communication reports on the high performance of hydrogen-terminated silicon nanowires and silicon nanowires coated with metal (Ag, Cu) nanostructures for the photodegradation of rhodamine B under UV and visible light irradiation.     

Combinatorial methods and recurrence relations with two indices

In this paper, we solve several recurrence relations with two indices by using combinatorial methods. Moreover, we present several recurrence relations with two indices related to Dyck paths and Schröder paths.