A faster algorithm for computing a longest common increasing subsequence

Let $A=\langle a_1,\dots,a_n\rangle$ and $B=\langle b_1,\dots,b_m \rangle$ be two sequences with $m \ge n$, whose elements are drawn from a totally ordered set. We present an algorithm that finds a longest common increasing subsequence of $A$ and $B$ in $O(m\log m+n\ell\log n)$ time and $O(m + n\ell)$ space, where $\ell$ is the length of the output. A previous algorithm by Yang et al. needs $\Theta(mn)$ time and space, so ours is faster for a wide range of values of $m,n$ and $\ell$

Reachability substitutes for planar digraphs

Given a digraph $G = (V,E)$ with a set $U$ of vertices marked ``interesting,'' we want to find a smaller digraph \RS{} = (V',E') with $V' \supseteq U$ in such a way that the reachabilities amongst those interesting vertices in $G$ and \RS{} are the same. So with respect to the reachability relations within $U$, the digraph \RS{} is a substitute for $G$. We show that while almost all graphs do not have reachability substitutes smaller than \Ohmega(|U|^2/\log |U|), every planar graph has a reachability substitute of size \Oh(|U| \log^2 |U|). Our result rests on two new structural results for planar dags, a separation procedure and a reachability theorem, which might be of independent interest

Group Theoretical Approach To Squeezed States Using Generalized Bose Operators

Generalized, k-boson Holstein-Primakoff realizations of SU(2) and SU(1,1) are introduced In terms of generalized bose operators. The corresponding group theoretical coherent states are studied with respect to their squeezing properties relative to k-boson dynamical variables

Generalized Holstein-Primakoff Squeezed States for SU(n)

We show how to define multi-photon, many-mode squeezed states for SU(n), using a generalized Holstein-Primakoff realization. We prove that for the class of realizations given, the resulting squeezing reduces to that of SU(2), and exemplify with a specific calculation for SU(3)

Dobiński relations and ordering of boson operators

We introduce a generalization of the Dobiński relation, through which we define a family of Bell-type numbers and polynomials. Such generalized Dobiński relations are coherent state matrix elements of expressions involving boson ladder operators. This may be used in order to obtain normally ordered forms of polynomials in creation and annihilation operators, both if the latter satisfy canonical and deformed commutation relations

Exponential Operators, Dobinski Relations and Summability

We investigate properties of exponential operators preserving the particle number using combinatorial methods developed in order to solve the boson normal ordering problem. In particular, we apply generalized Dobinski relations and methods of multivariate Bell polynomials which enable us to understand the meaning of perturbation-like expansions of exponential operators. Such expansions, obtained as formal power series, are everywhere divergent but the Pade summation method is shown to give results which very well agree with exact solutions got for simplified quantum models of the one mode bosonic systems.Comment: Presented at XIIth Central European Workshop on Quantum Optics, Bilkent University, Ankara, Turkey, 6-10 June 2005. 4 figures, 6 pages, 10 reference

Monomiality principle, Sheffer-type polynomials and the normal ordering problem

We solve the boson normal ordering problem for $(q(a^\dag)a+v(a^\dag))^n$ with arbitrary functions $q(x)$ and $v(x)$ and integer $n$, where $a$ and $a^\dag$ are boson annihilation and creation operators, satisfying $[a,a^\dag]=1$. This consequently provides the solution for the exponential $e^{\lambda(q(a^\dag)a+v(a^\dag))}$ generalizing the shift operator. In the course of these considerations we define and explore the monomiality principle and find its representations. We exploit the properties of Sheffer-type polynomials which constitute the inherent structure of this problem. In the end we give some examples illustrating the utility of the method and point out the relation to combinatorial structures.Comment: Presented at the 8'th International School of Theoretical Physics "Symmetry and Structural Properties of Condensed Matter " (SSPCM 2005), Myczkowce, Poland. 13 pages, 31 reference

Geometric approach to nonlinear coherent states using the Higgs model for harmonic oscillator

In this paper, we investigate the relation between the curvature of the physical space and the deformation function of the deformed oscillator algebra using non-linear coherent states approach. For this purpose, we study two-dimensional harmonic oscillators on the flat surface and on a sphere by applying the Higgs modell. With the use of their algebras, we show that the two-dimensional oscillator algebra on a surface can be considered as a deformed one-dimensional oscillator algebra where the effect of the curvature of the surface is appeared as a deformation function. We also show that the curvature of the physical space plays the role of deformation parameter. Then we construct the associated coherent states on the flat surface and on a sphere and compare their quantum statistical properties, including quadrature squeezing and antibunching effect.Comment: 12 pages, 7 figs. To be appeared in J. Phys.

Combinatorial Solutions to Normal Ordering of Bosons

We present a combinatorial method of constructing solutions to the normal ordering of boson operators. Generalizations of standard combinatorial notions - the Stirling and Bell numbers, Bell polynomials and Dobinski relations - lead to calculational tools which allow to find explicitly normally ordered forms for a large class of operator functions.Comment: Presented at 14th Int. Colloquium on Integrable Systems, Prague, Czech Republic, 16-18 June 2005. 6 pages, 11 reference