The linear and nonlinear optical (NLO) properties of a series ?-shaped derivatives containing 4,5-dicyanopyrazine acceptor unit, N,N-dimethylamino donor groups systematically enlarged ?-conjugated linkers are investigated by means UV/Visible Hyper-Rayleigh scattering spectroscopies. Density functional theory calculations also carried out to rationalize the magnitude symmetry NLO responses. resu...

Lauren Williams (joint work with Einar Steingrímsson) We introduce and study a class of tableaux which we call permutation tableaux; these tableaux are naturally in bijection with permutations, and they are a distinguished subset of the " Le-diagrams " of Alex Postnikov. The structure of these tableaux is in some ways more transparent than the structure of permutations; therefore we believe tha...

The pattern discovery task (or equivalently motif inference) is the knowledge discovery process that, given a dataset and some constrains either on the combinatorial pattern structure or on the occurrence lists, returns all the patterns satisfying the given constraints. In this thesis, we consider the problem of discovering patterns in sequential data, such as texts, biological sequences, acces...

We consider the function m[k](q) that counts the number of cycle permutations of a finite field Fq of fixed length k such that their permutation polynomial has the smallest possible degree. We prove the upper–bound m[k](q) ≤ (k−1)!(q(q−1))/k for char(Fq) > e(k−3)/e and the lower–bound m[k](q) ≥ φ(k)(q(q−1))/k for q ≡ 1 (mod k). This is done by establishing a connection with the Fq–solutions of ...

Let H be a subgroup of the multiplicative group of a finite field. In this note we give a method for constructing permutation polynomials over the field using a bijective map from H to a coset of H. A similar, but inequivalent, method for lifting permutation behaviour of a polynomial to an extension field is also given.

‎In this work‎, ‎a regular polygon with $n$ sides is described by a periodic (circular) sequence with period $n$‎. ‎Each element of the sequence represents a vertex of the polygon‎. ‎Each symmetry of the polygon is the rotation of the polygon around the center-point and/or flipping around a symmetry axis‎. ‎Here each symmetry is considered as a system that takes an input circular sequence and g...

We consider rational functions of the form V(x)/U(x), where both V(x) and U(x) are relatively prime polynomials over finite field \(\mathbb {F}_q\). Polynomials that permute elements a field, called permutation (PPs), have been subject research for decades. Let \({\mathcal {P}}^1(\mathbb {F}_q)\) denote {F}_q\cup \{\infty \}\). If function, permutes {F}_q)\), it is function (PRF). \(N_d(q)\) nu...