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

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

Heisenberg-Weyl algebra revisited: Combinatorics of words and paths

The Heisenberg-Weyl algebra, which underlies virtually all physical representations of Quantum Theory, is considered from the combinatorial point of view. We provide a concrete model of the algebra in terms of paths on a lattice with some decomposition rules. We also discuss the rook problem on the associated Ferrers board; this is related to the calculus in the normally ordered basis. From this starting point we explore a combinatorial underpinning of the Heisenberg-Weyl algebra, which offers novel perspectives, methods and applications.Comment: 5 pages, 3 figure

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

Zinc differentially modulates DNA damage induced by anthracyclines in normal and cancer cells

Zinc is one of the most essential trace elements in human organism. Low blood level of zinc is often noted in acute lymphocytic leukemia (ALL). Treatment with zinc adjuvant is hypothesized to accelerate recovery from ALL, and in conjunction with chemotherapy, cure ALL. Aim: We determined the effect of zinc on DNA damage induced by doxorubicin and idarubicin, two anthracyclines used in ALL treatment. Methods: The experiment was performed on acute lymphoblastic leukemia cell line (CCRF-CEM) and lymphocytes from peripheral blood of healthy, adult subjects. To evaluate the level of DNA damage the comet assay in the alkaline version was used. Results: We observed a significant difference in DNA damage level between normal and cancer cells in the presence of zinc. Cancer cells exhibited a significant increase of DNA damage in the presence of zinc, while in lymphocytes no such effect was observed. Conclusion: Our results suggest that zinc may protect normal cells against DNA-damaging action of anthracyclins and increase this action in cancer cells

Combinatorics and Boson normal ordering: A gentle introduction

We discuss a general combinatorial framework for operator ordering problems by applying it to the normal ordering of the powers and exponential of the boson number operator. The solution of the problem is given in terms of Bell and Stirling numbers enumerating partitions of a set. This framework reveals several inherent relations between ordering problems and combinatorial objects, and displays the analytical background to Wick's theorem. The methodology can be straightforwardly generalized from the simple example given herein to a wide class of operators.Comment: 8 pages, 1 figur

Amifostine can differentially modulate DNA double-strand breaks and apoptosis induced by idarubicin in normal and cancer cells

We have previously shown that amifostine differentially modulated the DNA-damaging action of idarubicin in normal and cancer cells and that the presence of p53 protein and oncogenic tyrosine kinases might play a role in this diversity. Aim: To investigate further this effect we have studied the influence of amifostine on idarubicin-induced DNA double-strand breaks (DSBs) and apoptosis. Methods: We employed pulse-field gel electrophoresis () for the detection of DSBs and assessment of their repair in human normal lymphocytes and chronic myelogenous leukaemia K562 cells lacking p53 activity and expressing the BCR/ABL tyrosine kinase. Apoptosis was evaluated by caspase-3 activity assay assisted by the alkaline comet assay and DAPI staining. Results: Idarubicin induced DSBs in a dose-independent manner in normal and cancer cells. Both types of the cells did not repair these lesions in 120 min and amifostine differentially modulated their level — decreased it in the lymphocytes and increased in K562 cells. In contrast to control cells, amifostine potentated apoptotic DNA fragmentation, chromatin condensation and the activity of caspase-3 in leukaemia cells. Conclusion: Amifostine can differentially modulate DSBs and apoptosis induced by idarubicin in normal and cancer cells. It can protect normal cells against drug-induced DNA damage and it can potentate the action of the drug in leukaemic cells. Further studies on link between amifostine-induced modulation of DSBs and apoptosis of cancer cells will bring a deeper insight into molecular mechanism of amifostine action.Ранее нами было показано, что амифостин дифференциально модулирует ДНК-повреждающее действие идарубицина в нормальных и злокачественных клетках, и что наличие белка р53 и онкогенных тирозин киназ может иметь значение для этих различий. Цель: изучить влияние амифостина на идарубицин-опосредованные двухнитевые разрывы ДНК (DSBs) и апоптоз. Методы: мы применили гель-электрофорез в пульсирующем поле (PFGE) для выявления DSBs и изучения их репарации в нормальных лимфоцитах человека и клетках K562 хронической миелоидной лейкемии, у которых р53 неактивен и экспрессирована BCR/ABL-тирозин киназа. Апоптоз оценивали с помощью реактивов для выявления активности каспазы-3, проведения щелочного гель-электрофореза одиночных клеток и DAPI-окрашивания. Результаты: идарубицин вызывает образование DSBs в нормальных и злокачественных клетках независимо от дозы. Оба типа клеток не репарировали эти повреждения за 120 мин, при этом амифостин дифференциально молулировал уровень DSBs — уменьшал в лимфоцитах и увеличивал в K562-клетках. В отличие от контрольных клеток амифостин потенциировал апоптическую фрагментацию ДНК, конденсацию хроматина и активность каспазы-3 в лейкемических клетках. Выводы: амифостин может дифференциально модулировать DSBs и апоптоз, вызванные идарубицином в нормальных и злокачественных клетках. Он может защитить нормальные клетки от повреждения ДНК, вызванного химиопрепаратом, и в то же время потенциировать действие препарата на лейкемические клетки. Дальнейшие исследования связи между вызванной амифостином модуляцией DSBs и апоптоза опухолевых клеток позволит лучше понять молекулярные механизмы действия амифостина

Hierarchical Dobinski-type relations via substitution and the moment problem

We consider the transformation properties of integer sequences arising from the normal ordering of exponentiated boson ([a,a*]=1) monomials of the form exp(x (a*)^r a), r=1,2,..., under the composition of their exponential generating functions (egf). They turn out to be of Sheffer-type. We demonstrate that two key properties of these sequences remain preserved under substitutional composition: (a)the property of being the solution of the Stieltjes moment problem; and (b) the representation of these sequences through infinite series (Dobinski-type relations). We present a number of examples of such composition satisfying properties (a) and (b). We obtain new Dobinski-type formulas and solve the associated moment problem for several hierarchically defined combinatorial families of sequences.Comment: 14 pages, 31 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

Quantum-like dynamics applied to cognition: a consideration of available options

Quantum probability theory (QPT) has provided a novel, rich mathematical framework for cognitive modelling, especially for situations which appear paradoxical from classical perspectives. This work concerns the dynamical aspects of QPT, as relevant to cognitive modelling. We aspire to shed light on how the mind's driving potentials (encoded in Hamiltonian and Lindbladian operators) impact the evolution of a mental state. Some existing QPT cognitive models do employ dynamical aspects when considering how a mental state changes with time, but it is often the case that several simplifying assumptions are introduced. What kind of modelling flexibility does QPT dynamics offer without any simplifying assumptions and is it likely that such flexibility will be relevant in cognitive modelling? We consider a series of nested QPT dynamical models, constructed with a view to accommodate results from a simple, hypothetical experimental paradigm on decision-making. We consider Hamiltonians more complex than the ones which have traditionally been employed with a view to explore the putative explanatory value of this additional complexity. We then proceed to compare simple models with extensions regarding both the initial state (e.g. a mixed state with a specific orthogonal decomposition; a general mixed state) and the dynamics (by introducing Hamiltonians which destroy the separability of the initial structure and by considering an open-system extension). We illustrate the relations between these models mathematically and numerically. This article is part of the themed issue 'Second quantum revolution: foundational questions'